## 2. Temporal Locality

#### 2.1. Definition

In seeking to set out a definition of temporal locality, a natural starting point is the standard mathematical definition of spatial locality [

2,

3]:

**Definition** **1.** **Spatial Locality:** Suppose that two observers, Alice and Bob, perform measurements on a shared physical system: Alice performs a measurement with setting a and obtains a measurement outcome A, while Bob performs a measurement with measurement setting b and obtains a measurement outcome B. Let λ be the joint state of the shared system prior to the two measurements. Then:We can straightforwardly apply this language to the temporal case:

**Definition** **2.** **Temporal Locality:** Suppose that two observers, Alice and Bob, perform measurements on a shared physical system. At some time ${t}_{a}$, Alice performs a measurement with measurement setting a and at some time ${t}_{a}+\delta $ she obtains a measurement outcome A; likewise, at some time ${t}_{b}$, Bob performs a measurement with measurement setting b and at some time ${t}_{b}+\delta $ he obtains a measurement outcome B. Let $\lambda \left({t}_{a}\right)$ be the state of the world at time ${t}_{a}$ and let $\lambda \left({t}_{b}\right)$ be the state of the world at time ${t}_{b}$. Then:The central idea of this definition is that in a temporally local world there would be “no action at a temporal distance”, i.e., all influences on a measurement outcome would be mediated by the state of the world immediately prior to the measurement. Of course, the definition does not lead to any specific theoretical constraints without some specification of what is included in “the state of the world at time t”, but in this article we will not single out any unique way of characterising this state: instead, we will set out a range of options, acknowledging that there are a number of related concepts floating around in modern physics which might reasonably be subsumed under the heading of temporal locality.

It is helpful to approach this range of possibilities by describing some different ways in which physics might fail to be temporally nonlocal. First, a theory might fail to be temporally local by postulating non-Markovian laws, meaning that the results of a measurement at a given time can depend on facts about earlier times even if there is no record of those facts in the state of the world immediately prior to the measurement. Note that this is possible only within a theory in which the state of the world at time t, if such a thing exists, does not always contain complete information about everything that has happened before t. Alternatively, a theory might fail to be temporally local by being retrocausal, meaning that the results of measurements at a given time may depend in part on information about the future. We reinforce that retrocausality does not immediately imply temporal nonlocality: a retrocausal theory is temporally nonlocal only if it tells us that the result of a measurement can depend on facts about the future even if there is no record of those facts in the state of the world immediately prior to the measurement. Therefore this type of temporal nonlocality is possible only within a theory in which the state of the world at time t, if such a thing exists, does not always contain complete information about everything that happens after t—in particular, it must not be the case that the state of the world immediately prior to the measurement already contains a record of the future outcome of the measurement, as for example in theories which are deterministic in the traditional sense, meaning that the state of the world at a given time determines everything that happens at later times. Finally, a theory might fail to be temporally local by being atemporal, meaning that the course of history is determined “all at once” by external, global laws of nature, in much the same way as the rules of the game of sudoku apply to the whole grid at once rather than dictating the entries column by column from left to right. In such a theory, the result of a measurement at a given time may depend on global facts even if there is no record of those facts in the state of the world immediately prior to the measurement, and thus an atemporal theory will usually be temporally nonlocal, unless of course the theory tells us that the state of the world at time t always contains complete information about the history of the entire universe. Each of these alternatives singles out a different sense of temporal (non-)locality, and all three raise interesting possibilities for new ways of thinking about physics.

#### 2.2. Motivation

Although physicists are certainly aware that the assumption of temporal locality is problematic, as a methodological principle it remains very widespread in the field. Although, presumably, some physicists would fight to the death for temporal locality, it seems likely that many others retain it simply because they regard it as a harmless simplification. However, we argue that the assumption is by no means harmless: temporal locality is deeply woven into many of the key results on which our present understanding of the interpretation of quantum theory is founded, and unpicking it would require a radical reinterpretation of the significance of those results.

In particular, much recent work in quantum foundations has been based within the “ontological models” framework introduced by Spekkens in [

4], where it is supposed that every system has a single real “ontic state”, which determines the probabilities for the outcomes of any measurement on that system. An ontological model thus consists of a space

$\mathsf{\Lambda}$ of ontic states

$\lambda $, a set of probability distributions

${\mu}^{P}\left(\lambda \right)$ giving the probability that the system ends up in the state

$\lambda $ when we perform the preparation procedure

P, a set of response functions

${\overrightarrow{\xi}}_{M,O}\left(\lambda \right)$ giving the probability that we obtain outcome

O when we perform measurement

M on a system whose ontic state is

$\lambda $, and a set of column-stochastic matrices

${T}^{X}$ representing the way in which the ontic state is transformed when some operation

X is applied to the system. Note that talk of “ontic states” does not imply that we are postulating the existence of hidden variables, because the “ontic state” could simply be the quantum state [

5]. It should also be reinforced that one can make use of the formalism of ontological models without necessarily interpreting it as an attempt at a faithful representation of reality—Spekkens himself prefers to regard it as a classification schema which enables us to give precise mathematical definitions for concepts like contextuality [

6]. Nonetheless, it seems to be the case that this formalism, or something close to it,

is often regarded as a description of reality, and indeed as the only possible way of describing reality—for example, in [

7], it is claimed that any model in which correlations are not explained by appeal to ontic states should not really be regarded as a realist model at all.

The ubiquity of this method of analysis matters, because the ontological models framework is explicitly temporally local. Not only that, temporal locality is the founding principle of the approach: the entire project of constructing an ontological model is premised on the assumption that measurement results must depend only on the information available in the ontic state at the time of the measurement. Consequently, temporal locality is the keystone of a number of influential results parsed in the language of ontological models, such as Spekkens’ generalized proofs of contextuality [

4], the Colbeck–Renner theorem [

8], Hardy’s theorem [

9], and the Pusey–Barrett–Rudolph (PBR) theorem [

10].

As a case study, let us consider the PBR theorem, which states that no model in which the quantum state is not an “element of reality” can reproduce all the quantitative predictions of quantum mechanics. Now, the term “element of reality” is a reference to a definition set out by Harrigan and Spekkens [

11], but although this definition refers only to instantaneous facts, the proof of the PBR theorem depends implicitly on assumptions not only about states at a given time, but about the persistence of those states over time: PBR write that if there exists a set of four preparation procedures which all have some probability of preparing the same ontic state, then when this state is prepared, “

the measuring device is uncertain which of the four possible preparation methods was used, and on these occasions it runs the risk of giving an outcome that quantum theory predicts should occur with probability 0” [

10]. This makes it clear that the argument also requires the assumption that the outcome of the measurement can depend on facts about the system’s preparation only via the mediation of an intervening state, so the PBR theorem should really be glossed as follows:

either the quantum state is ontological,

or some quantum measurement results must depend in a temporally nonlocal way on events at other times. In this context, then, the assumption of temporal locality is decidedly non-trivial—for example, anyone who wishes to push back against the ontological picture of quantum states should certainly be raising questions about this assumption.

Moreoever, most mainstream interpretations of quantum mechanics, including the Everett interpretation, spontaneous collapse models and the de Broglie Bohm approach, are prima facie temporally local. (We do not mean to suggest that these models could not be phrased in a temporally nonlocal way, nor even to assert that this has not already been done somewhere in the literature, but it does seem to be the case that temporal nonlocality is not a central feature of any of these interpretations). This suggests that fully embracing temporal nonlocality might open up untapped possibilities for the interpretation of quantum theory, and hence the whole landscape of quantum foundations becomes markedly different when temporal nonlocality is taken seriously.

## 3. Origins

Given that temporal locality plays such a key role in our modern understanding of quantum theory, it is important to understand the intellectual history of this idea. In this section, we argue that a number of historical and psychological factors are likely to have contributed to its prominence; indeed, temporal locality is, in a sense, built into the very structure of physics. Consider the long tradition of presenting theories in terms of their kinematics (the space of physical states postulated by the theory) and their dynamics (the set of laws by which these states evolve, according to the theory). This distinction can be traced back at least to Newton, who may have been the first to make a clear distinction between laws and initial conditions [

12], and since Newton’s time the formulation has become widespread: it is now almost mandatory to introduce a new physical theory by setting out a space of physical states and a set of evolutionary laws [

4]. However, a physical state is, almost by definition, that which carries information from one time to another by means of its dynamical evolution, and thus by employing this mode of presentation we are already very close to assuming that information about one time can influence the results of measurements at other times only via a mediating physical state, thus ruling out temporal nonlocality almost by fiat. Temporal locality is thus very deeply ingrained in the way physicists are taught to think about physics.

There are also straightforward pragmatic reasons why temporal locality should have gained such ascendancy in science. After all, we ourselves are local agents—if we wish to influence events at a spatial or temporal distance we must do so via some spatiotemporally continuous process of mediation—and the fact that these constraints are, for us, so immediate and insurmountable naturally leads us to imagine that the laws of nature must be subject to similar constraints. The empirical results of quantum mechanics, such as the violation of Bell’s inequality, have give us convincing reasons to question the resulting attachment to spatial locality, but temporal nonlocality has not thus far been subject to the same level of analysis and hence lives on in the ways we think and talk about quantum mechanics. Furthermore, as scientists, our primary practical interest is in formulating laws which enable us to predict the future given our knowledge of the present state of the world, and it is easy to move from the fact that most of the laws proposed by physicists have this form to the conclusion that the true underlying laws of nature must take the same form. However, it would be naive to suppose that the true laws of nature look exactly like the type of laws that human agents are most interested in formulating: as Wharton puts it: “There’s one last anthropocentric attitude that needs to go, the idea that the computations we perform are the same computations performed by the universe”. Assuming that our point of view is not central to the universe, it would be highly suspicious if the laws of nature were to be arranged so conveniently for us.

It also seems likely that certain elements of temporal locality have their origin in the viewpoint known in academic philosophy as “presentism”, which holds that the only things which are real are the things which exist now [

13,

14,

15]. A realist about science will clearly want to insist that measurement results can depend only on things that are real, and hence a realist who subscribes to presentism is compelled to believe that measurement results can depend only on facts about the world immediately prior to the measurement. Presentism is a very old philosophical idea, appearing in the writings of Aristotle and St. Augustine, and playing an important role in Buddhist philosphy, although with the advent of special relativity it has gone somewhat out of vogue as an explicit philosophical thesis: much has been written on the question of whether or not relativity makes presentism untenable [

16,

17,

18,

19], but whether or not the two can be formally reconciled, they are certainly in tension with one another. Nonetheless, although there are few modern physicists who would self-describe as presentists, the intuitive picture of the present as somehow specially privileged remains hard to shake, and it is likely that some element of this way of thinking contributes to the general conviction that scientific theories should respect temporal locality.

We reinforce that although these historical and psychological observations go some way towards explaining why our scientific theories tend to be temporally local, they do not offer any epistemic justification for thinking that the world actually is temporally local. Of course, it may be the case that some epistemic justification can be provided, but if such a justification exists it is certainly not commonly known and hence cannot be regarded as the main reason why our theories exhibit this feature. This indicates that the prominence of temporal locality in our standard approaches to physics may not be entirely rational and perhaps deserves greater scrutiny than it has thus far received.

#### The Pragmatic Argument

At this juncture, a defender of temporal locality might wish to suggest a different type of justification, using pragmatic rather than epistemic arguments. In particular, one might worry that if we accept that events at this moment may depend on events at any point in the past or future, it will become very difficult to track all the variables which might be relevant to the outcome of an experiment, and the whole scientific enterprise will be under threat. Indeed, similar objections were raised by Einstein concerning

spatial nonlocality [

20]:

An essential aspect of this arrangement of things in physics is that they lay claim, at a certain time, to an existence independent of one another, provided these objects “are situated in different parts of space”. Unless one makes this kind of assumption about the independence of the existence (the “being-thus”) of objects which are far apart from one another in space ... physical thinking in the familiar sense would not be possible. It is also hard to see any way of formulating and testing the laws of physics unless one makes a clear distinction of this kind.

However, despite Einstein’s concerns, it has not proven to be impossible to formulate a theory which allows for spatial nonlocality, because the nonlocal relations between events are governed by laws which enable us to identify regularities in patterns of dependence even between spatially separated events. Likewise, in principle it would not be impossible to move forward with a theory which allows for temporal nonlocality, provided that events at a time depend on events at other times in some regular, formalisable way—indeed, we already have a way of tracking patterns of dependence both temporally and spatially, since the quantum state gives a concise summary of all the information about the history of a system which we know to be relevant to the results of future measurements performed on that system. Therefore the assumption of temporal locality is not forced upon us by practical considerations, and it behoves us to consider the possibility that an explicitly temporally nonlocal theory might enable us to identify and track further regularities.

## 6. Dynamics and Kinematics

We earlier identified the distinction between dynamics and kinematics as an important contribution to the status of temporal locality in physics. In this section we review recent work on this subject and discuss some resulting insights for the status of temporal locality in modern physics.

#### 6.1. Spekkens on Dynamics vs. Kinematics

Spekkens has singled out the distinction between kinematics and dynamics as a potentially problematic feature of our standard physical paradigms: in [

58], he argues that when new experimental data appears to falsify our existing theory, we can always choose freely whether to respond by altering the kinematics or the dynamics, and he gives a number of illustrative examples. He thus concludes that the distinction between kinematics and dynamics is doing no explanatory work in our theories, and appeals to ontological parsimony to motivate his call for physicists to move past this particular paradigm.

While we concur that the kinematics/dynamics split is problematic, we note that care must be taken with this line of argument to avoid slipping into conventionalism about the whole of science. Spekkens asserts that his approach “does not force us to operationalism”, because he advocates only the rejection of distinctions which we can freely transform away without changing empirical predictions, and which are therefore not doing explanatory work. However, as Quine has shown, it can reasonably be argued that physical theories have empirical consequences only taken as a whole, and that consequently we always have freedom to choose which element of a theory to change in response to new empirical evidence, which would suggest that by Spekkens’ criterion no distinction in any scientific theory is doing explanatory work [

59,

60]. Thus Spekkens’ line of argument would seem to lead to the conclusion that we should simply give up on trying to formulate theories whose ontologies are endowed with nontrivial structure, a conclusion which realists about science will surely wish to avoid.

To do so, we must understand why Spekkens’ argument has particular relevance in the context of the kinematical-dynamical distinction. In particular, let us reinforce that the distinction between kinematics and dynamics is not simply an individual element of some specific theory; the fact that it has become de rigueur to present new theories in this framework has made the kinematical-dynamical split into a meta-principle which physicists educated in this tradition may well regard as a defining feature of any meaningful scientific theory. By pointing out that the kinematical-dynamical distinction is not forced on us by any empirical evidence, Spekkens demotes it from a meta-principle back to a specific ontological hypothesis which should be subject to the same scrutiny and criticism as any other ontological hypothesis. The argument can then be understood as follows: as realists we choose to attach credence to certain ontologies, despite underdetermination by the empirical evidence, on the grounds of theoretical virtues like simplicity and explanatory power, and the same sorts of assessments should be applied to the distinction between kinematics and dynamics. Since theoretical distinctions in general do not have empirical content in and of themselves, it is no good insisting on a distinction between kinematics and dynamics in advance of specifying a particular ontology: we must evaluate the theoretical virtues of complete ontologies, some of which may incorporate such a split, others of which may not.

For clarity, at this point, we must mention a different way of thinking about the kinematics/dynamics distinction that has arisen through recent work on the philosophy of special relativity. In this tradition, “what it means for a phenomenon to be kinematical ... is that it is nothing but a specific instance of some generic feature of the world ... (and that) there is nothing more to learn from that particular phenomenon, neither about the specific system in which it occurs nor about the generic feature it instantiates”. In other words, the dynamics/kinematics distinction is regarded as a stipulation about which things need to be explained and which things can be taken for granted. This is clearly quite a different concept from the notion of kinematics and dynamics that we have thus far referred to in this article, and to which Spekkens’ argument pertains. Certainly it presupposes much less—in particular, such a distinction would still be perfectly meaningful within a theory which does not postulate a space of states and a set of evolutionary laws, whereas the more traditional way of distinguishing between kinematics and dynamics would be inapplicable in such a case. Nonetheless, we conjecture that similar arguments can be made about this more general distinction. Brown points out that “the distinction between kinematics and dynamics is not fundamental” and cites Pauli as making the same point in 1921, and, again, once this point is accepted it seems unreasonable to demand that all new theories must be presented in the framework of kinematics and dynamics: we may well find it heuristically useful to employ such a distinction in any particular case, but the judgement of its usefulness must be made in context, not in advance of the specification of a theory. We leave a more detailed development of this line of argument to future work.

#### 6.2. Example: Causal Set Theory

As an example of a theory in which the kinematics/dynamics distinction may be less useful, consider the case of causal set theory, an approach to quantum gravity which holds that spacetime is fundamentally discrete. The “state space” of this theory is the space of causal sets—that is, sets of spacetime events with a partial order event defined over them. A causal set is, essentially, an entire history of a universe, with time and space being emergent from the partial order between pointlike events. However, it is not sufficient for the theory to simply specify this kinematical state space, because without further restrictions we will find that the majority of causal sets do not give rise to any low-dimensional emergent spacetime (a spacetime is said to emerge from a causal set iff it faithfully approximates the causal set—that is, we can embed the causal set into the spacetime in such a way that the causal relations are preserved (x lies before y in the partial order iff the embedding of x is in the past lightcone of the embedding of y), and on average one element of the causal set is mapped onto each Planck-sized volume of the spacetime, and the spacetime does not have structure at scales below the mean spacing of the events) so we must add in some way of singling out the permissible causal sets.

The standard way of doing this is to impose “dynamics”, such as the classical sequential growth model in which elements are probabilistically added to the set one by one [

61]. Proponents of causal set theory like to advertise it as an advantage of their approach that this dynamics provides us with a relativistically covariant notion of “becoming”, allowing us to rescue the notion of temporal becoming and hence salvage our intuitive notion of time [

62]. However, this claim cannot quite be taken at face value, because we encounter a difficulty akin to Smart’s objection to the A-theory of time. Smart famously pointed out that if time really passes, we ought to be able to specify the rate at which it passes, which would require a second time-dimension with respect to which the passage of ordinary time can be measured [

63]; and likewise in causal set theory, talking about the growth of the causal set seems to presuppose an external time dimension in which this growth can take place. The difficulty is all the more pertinent since the founding principle of the theory is that spacetime supervenes on the causal set [

64] and thus proposing a dimension of time external to the causal set would seem to undermine the whole project. Rideout and Sorkin attempt to get around this by arguing that the birthing of events should be regarded as “constituting” time rather than occurring in time [

61], but this seems like overkill: since spacetime supervenes on the causal set, a complete causal set already “constitutes” time, without any need to add in a process of growth. Furthermore, to ensure that the growth process satisfies general covariance, it is necessary to impose the requirement of discrete general covariance on the dyamics, meaning that the probability of reaching a particular final causet is independent of the path taken to reach that final causet—i.e., the probability does not depend on the order in which the elemets of the causet were “birthed”. It is standard to interpret this by saying that there is no fact of the matter about which path was taken—the choice of path is pure gauge [

61]—but this makes it implausible to regard the growth of the causal set as a real physical process, since probabilities are ultimately attached to the causal sets themselves rather than to the transitions that occur during the supposed growth [

65,

66]. Wütrick and Callender argue that these considerations simply show that modern physics requires us to adopt a ‘novel and exotic’ notion of becoming in which we are generally prohibited from saying which elements of the causal set exist at any stage of its growth [

66]. However, this novel notion of “becoming” is so far removed from our intuitive notion of becoming that it is doubtful whether it can really be said to salvage our intuitive notion of time; moreoever, given that the dynamics cannot be taken literally, there is prima facie no way in which the growth of the causal set could even serve to explain why we have the subjective experience of becoming. The growth model, in fact, does not seem to add anything to the theory in terms of explanatory power: insofar as the dyamics succeeds in explaining why certain causal sets are permissible while others are not, and/or why the world is constituted by one causal set rather than another, the real explanatory work is done by the final probability distribution over causal sets rather than by the process of growth.

Thus it seems that all the growth picture is really doing is making causal set theory subjectively more palatable by soothing our uneasiness about attaching probabilities to entire courses of history, and, of course, allowing the causal set theorists to express their theory in the traditional framework of ‘kinematics vs. dynamics”. Thus, although it is possible to make a distinction between kinematics and dynamics within causal set theory, in this particular case the distinction seems not to be very useful and may in fact be holding us back from understanding the theory properly: perhaps the causal set theorists would do better to embrace the global nature of their theory and explicitly attach probabilities to entire causal sets, retaining the “growth” dynamics only as a calculational tool or perhaps even getting rid of it entirely in favour of a different way of calculating the relevant probabilties.

We have singled out the causal set approach here because the awkwardness of attempting to distinguish between kinematics and dynamics is particularly clear in this context, but we would contend that similar points apply more generally. Theories should not be forced into the kinematics-dynamics framework if they are not a natural fit for that framework: this practice imposes an artifical form of temporal locality on theories which are not inherently temporally local in their mathematical structure, which is likely to impede both understanding and also further theoretical progress.

## 7. Temporal Bell Inequalities and Entanglement in Time

The stark differences between contemporary attitudes to spatial and temporal locality can largely be traced back to the existence of Bell’s inequalities and the fact that quantum mechanics is known to violate them [

67], an experimentally verified fact which has led the physics community to at take seriously the possibility that spatially nonlocal processes may exist. Of course, the implication is not undisputed; although a number of experimental loopholes in Bell’s theorem have been closed in recent years [

68,

69,

70], there remain untested assumptions, such as the possibility that the choices of measurement on the two sides of the apparatus are not truly independent [

71]. Furthermore, proponents of the Everett interpretation claim their approach can account for the Bell statistics in a spatially local way, and antirealists can avoid spatial nonlocality simply by denying that there exists any underlying process, local or otherwise, which accounts for the measurement statistics. However, each of these ways around the conclusion of the theorem requires us to accept a fairly extreme proposition of one type or another, so it is fair to say that, conditional on a set of assumptions which seem very plausible to many people, the violation of Bell’s inequality does indeed imply the existence of spatial nonlocality.

Thus it is very natural to consider whether some analogous set of equations are violated in the temporal case. The first point to be made is that the derivation of Bell’s theorem assumes both spatial and temporal locality. If we relax the assumption of temporal locality, then we could say, for example, that the result of the measurement may depend directly on the state of the system being measured at times other than the time of measurement, including

future times: as we noted above, the proponents of retrocausality have used this possibility to explain the violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality via a local interaction which is mediated via the future [

44]. Thus, it is not really fair to say that we have better evidence for spatial nonlocality than temporal nonlocality: we have exactly the same evidence for both. However, physicists have largely chosen to respond to this evidence by discarding spatial locality and retaining temporal locality (or indeed by arguing that we can salvage both), and therefore to have convincing evidence that points specifically to temporal locality, we would need not just a temporal analogue of Bell’s inequalities, but a stronger result which shows that spatial nonlocality is not enough to explain the empirical results of quantum mechanics: the quantum world must be temporally nonlocal as well.

There exist several inequalities—mostly governing sequences of measurements performed on a single quantum system—which have been referred to as “temporal Bell’s inequalities”, and we will now consider whether any of them might be capable of providing the right sort of evidence. First, it should be clear that in the context of repeated measurements on a single quantum system, the assumption of temporal locality alone will not allow us to derive anything, since we can always choose to retain temporal locality by assuming that the entire history of a system is recorded in its present state. Thus, to obtain meaningful results, some further assumption p must be made, so we are never going to obtain a result stronger than “if this inequality is violated, then either $\neg p$, or quantum mechanics is not temporally local”. For this to provide a convincing argument in favour of temporal locality, p would need to be an assumption so plausible that many people would be willing to abandon temporal locality before abandoning p.

In the case of the Leggett–Garg inequalites, the additional assumption is “macrorealism”, which is the claim that a macroscopic object is at any given time in a definite ontic state and it is possible to determine which state it is in without changing the state or the subsequent system dynamics. Macrorealism is a strong assumption—too strong, in fact, for our purposes, because the only measurements referenced in the Leggett–Garg inequalities are measurements which reveal the definite ontic state of the system at the time of the measurement. As noted in

Section 5.2, it is reasonable to assume that the state of the world at time

t includes all the definite ontic states of all systems which exist at time

t, and thus by definition the measurements referenced in the Leggett–Garg inequalities are only allowed to depend on the state of the world at the time of the measurement, which makes temporal locality irrelevant: whether or not the world is temporally nonlocal in general, for this specific type of measurement there is no freedom for the measurement result to depend on anything other than the present state of the world. There exist later reformulations of the Leggett–Garg inequalities which replace macrorealism with a weaker assumption, but most of these reformulations retain the assumption of “operational eigenstate realism”, that is, the assumption that quantum systems are necessarily in states in which the quantity being measured has a definite value which is revealed deterministically by the measurement, and again this makes the assumption of temporal nonlocality irrelevant [

72]. A similar issue arises for the set of temporal Bell inequalities derived in [

73], and used to demonstrate the phenomenon of “entanglement in time”. Here the derivation depends on temporal locality and also “realism”, defined as the assumption that measurement results are determined by hidden properties that the particles carry prior to and independently of observation; one assumes that the state of the world at the time of the measurement would be expected to includ these “hidden properties” and, thus under this assumption measurement results can depend only on the state of the world at the time of the measurement, so the auxiliary assumption already implies temporal locality, in the sense in which we have used the term.

Thus, if we are to derive a temporal Bell’s inequality has something to say about temporal locality in particular, we should look for an assumption

p which does not itself imply temporal locality. A possible candidate is put forward in [

74]: here, the derivation of the inequalities is based on the assumption that the results of measurements on a system of dimension

d should be simulable by an ordered set of classical systems with no more than

${log}_{2}\left(d\right)$ bits of communication between any consecutive pair of systems. It is helpful to split this assumption into two parts: first, the total amount of information about its history that can be carried forward in time by a quantum state of fixed dimension is upper bounded by

${log}_{2}\left(d\right)$ bits; and second, the result of a measurement on the system is statistically independent of all information about its history which is not stored in its present state, i.e., the measurements in question are temporally local. Ref. [

74] use this assumption to derive a bound on the minimum dimension of a system which can solve a certain sort of sequential problem, and then show that the problem can be solved by quantum systems of dimension smaller than this bound, indicating that quantum mechanics does not satisfy their assumption. This is exactly the kind of result needed to provide an argument for temporal locality: if we find it sufficiently unpalatable to postulate that quantum states may carry information greater than

${log}_{2}\left(d\right)$ forward in time, we will have to conjecture instead that the later measurement results depend directly on earlier measurement settings and outcomes without being mediated via information carried forward in the state, and thus we may regard the violation of this inequality as a direct demonstration of temporal nonlocality at work in quantum mechanics, in the same way that a Bell experiment is a direct demonstration of spatial nonlocality at work in quantum mechanics. Admittedly, it may not be the case that there are many people who find the bound

${log}_{2}\left(d\right)$ more intuitively plausible than temporal locality, but at least the result seems to be of the right form.

#### The Problem of Records

It may seem that the existence of records of past measurement results must always stymie any attempt to use the violation of some inequality to prove that the world must be both spatially and temporally nonlocal, since even if we do make an assumption like that of [

74] to the effect that a given system can only carry a bounded amount of information forward in time, a proponent of temporal locality could always claim that a given result depends on the record of a given measurement result stored

elsewhere in the present state of the world, rather than directly on the past events constituting the measurement. After all, in practice such records are very difficult (perhaps impossible!) to erase, and in any case, if a past measurement result could be permanently erased so that no record of it existed in the state of the world at the time of the next measurement, then we would never be able to observe the violation of the relevant inequality, since we could never have all the necessary results available to be compared at the same time.

We suggest the best way of resolving this difficulty is to adopt a halfway position inspired by our discussion in subsection “The Pragmatic Argument”. The problem that we are facing can be understood as a particular instance of the general problem identified by Einstein: if spatial locality is simply abandoned wholesale, it becomes impossible to identify and control all the factors which might possibly influence the results of an experiment, and thus we lose the ability to draw meaningful conclusions from experimental results. Therefore, as noted in subsection “The Pragmatic Argument”, to make progress we must assume that there are limits on spatial nonlocality. The most straightforward approach is to assume that the world is only as spatially nonlocal as quantum mechanics says it is, because then, provided the system being measured is in a sufficiently pure state, we can justify the assumption that the result of the measurement is independent both of records stored elsewhere in the world and of the state of the observer’s brain. The resulting inequality will still be theory dependent, but at the very least the violation of such an inequality, assuming we are not willing to abandon the assumption that quantum states of dimension d may only carry ${log}_{2}\left(d\right)$ bits of information forward in time, would force us to say either that the world must be temporally nonlocal or it must be more spatially nonlocal than quantum mechanics currently suggests.

An alternative would be to assume that we need only worry about spatial and/or temporal entanglement when the systems concerned can be connected via some reasonably simple spatiotemporal path, as for example in the case of two entangled particles which have interacted locally at some point in their shared past. This would have the advantage of removing any dependence on the present formalism of quantum mechanics, which may be desirable given that we do not know how much of that formalism would survive the move to a temporally nonlocal context, but on the other hand to make the criterion precise we would likely need a reasonably concrete proposal for an alternative theory, and at present only toy models are available to us for this purpose.

## 8. Conclusions

There already exists a small body of interesting work examining the possibility of what might be interpreted as temporally nonlocal approaches to quantum theory, although most of it has not yet reached the mainstream. Wharton, advocating the view that “the universe (runs) not as a computer, but as a global four-dimensional problem that (is) solved all at once” [

29], has made progress with retrocausal models [

75,

76,

77]; the consistent histories approach offers an approach to formulating laws of nature which constrain entire histories rather than moment-by-moment evolution [

78,

79], although there are a number of significant conceptual difficulties to be resolved, not least the question of what the probabilities prescribed by the theory are probabilities for [

80]; and Ref. [

81] puts forward a theoretical model, in which “

one particle at N times is ... equivalent to N (entangled) particles at one time”, which, by emphasizing the parallel between spatial nonlocality and time-evolution, seems to lead naturally to a temporally nonlocal view. Similarly, there exist interpretations of quantum mechanics whose ontology consists entirely of pointlike events, such as the GRW flash ontology [

23,

24] or Kent’s solution to the Lorentzian quantum reality problem [

82], and one possible interpretation of these approaches would be to say that they have done away with the need for an ontic state as the carrier of information from the past to the future and hence should be regarded as temporally nonlocal.

This existing work is very promising, but we would argue that it does not go far enough. These approaches have been postulated as part of the project of interpreting the existing framework of quantum mechanics (and/or quantum field theory), and yet, once we accept that the universe may be generically nonlocal across both time and space, it becomes at least plausible that quantum theory as we know it is simply the local limit of a global theory which applies constraints across the whole of space and time. This means there is scope to be more ambitious: temporal nonlocality may ultimately point us not just to a new interpretation of quantum mechanics but to a new theory altogether.