Possible consequences for physics of the negative resolution of Tsirelson's problem

Abstract: In 2020, Ji et al. [arXiv:2001.04383 and Comm.~ACM 64}, 131 (2021)] provided a proof that the complexity classes $\text{MIP}^\ast$ and $\text{RE}$ are equivalent. This result implies a negative resolution of Tsirelson's problem, that is, $C_{qa}$ (the closure of the set of tensor product correlations) and $C_{qc}$ (the set of commuting correlations) can be separated by a hyperplane (that is, a Bell-like inequality). In particular, there are correlations produced by commuting measurements (a finite number of them and with a finite number of outcomes) on an infinite-dimensional quantum system which cannot be approximated by sequences of finite-dimensional tensor product correlations. Here, we point out that there are four logical possibilities of this result. Each possibility is interesting because it fundamentally challenges the nature of spacially separated systems in different ways. We list open problems for making progress for deciding which of the possibilities is correct.