In the last decade, there have been several attempts to study and classify physical theories according to the correlations which they allow to generate between distant observers, see Black Box Quantum Theory. In this direction, we proposed the principle of macroscopic locality . In a nutshell, this axiom demands that coarse-grained extensive measurements of natural macroscopic systems must admit a classical description.
Consider a standard experiment of bipartite nonlocality, conducted by two parties, Alice and Bob, see the Figure.
A pair of particles is generated, Alice and Bob subject them to interactions x,y and, finally, they impinge on detectors a,b with probability P(a,b|x,y). In a macroscopic version of this experiment, N pairs of particles are created, Alice and Bob interact with each particle beam in ways x,y, and different intensities are measured at each detector.
When the number of pairs tends to infinity, it can be shown that, if the particles follow the laws of quantum physics, the resulting probability density ρ(I1,…,J1,…|x,y) can be explained by a classical model. That is, two macroscopic observers would not need to invoke non-classical physical theories to make sense of this experiment. In contrast, when the beam is composed of particle pairs described by certain supraquantum distributions, such as the Popescu-Rohrlich box, the macroscopic observations ρ(I1,…,J1,…|x,y) defy the laws of classical physics. Macroscopic Locality demands the existence of a classical limit for this class of experiments, and hence the latter boxes are not supposed to appear in any reasonable physical theory.
The set of microscopic bipartite boxes P(a,b|x,y) satisfying Macroscopic Locality is slightly bigger than the quantum set. Hence, as far as Macroscopic Locality goes, there is room for physical theories beyond quantum mechanics. The Macroscopic Locality set was also the first non-trivial example of a set of correlations closed under wirings.
By wiring, or combining inputs and outputs of independent boxes, two distant parties can generate a new effective bipartite box, see the figure below.
Clearly, if the original boxes can be realized in a given physical theory, so can the final effective box. In other words: physical sets of boxes must be closed under wirings. The condition of being closed under wirings limits severely the geometry of the physical sets. For years, only a handful of closed sets were identified: the classical set, the quantum set, the set of boxes compatible with macroscopic locality and the set of boxes compatible with relativistic causality (also known as the non-signalling set). These sets had a “Russian-doll” structure, i.e., the classical set is contained in the quantum set, that is contained in the ML set, that is contained in the non-signalling set. The evidence therefore pointed that the framework of black boxes could only accommodate finitely many different physical theories and that those formed a hierarchy.
In  we showed that, on the contrary, there exist infinitely many sets closed under wirings. Moreover, some of them satisfy non-trivial inclusion relations!
As explained in , the last result implies that there exist information-theoretic physical principles which are unstable under composition. For any such principle Q, there exists a pair of boxes P1,P2 such that infinitely many copies of one or the other are compatible with Q, but no physical theory satisfying Q can contain both P1 and P2. Expressions such as “the set of all correlations satisfying the principle of Information Causality” (see below), widely used in foundations of physics, may therefore have no meaning at all. It is interesting to remark that Macroscopic Locality can be shown not to exhibit this pathology: any number of boxes satisfying Macroscopic Locality cannot be engineered to violate this physical principle.
Apart from Macroscopic Locality, there have been another four proposals to bound the nonlocality of any supraquantum theory: Non-trivial Communication Complexity, No Advantage for Nonlocal Computation, Information Causality and Local Orthogonality. Long after the publication of these works, it was still not clear whether any of these principles or a subset of them was strong enough to single out quantum nonlocality. This would have implied that, at least at the level of correlations, there exist no reasonable physical theories beyond quantum physics.
This state of affairs changed in 2014, when we identified a supra-quantum set of correlations that, with the possible exception of Information Causality, does not violate any of the aforementioned principles  (there is plenty of numerical evidence that suggests that it also satisfies Information Causality). We call it the almost quantum set. It emerges naturally in a consistent histories approach to theories beyond quantum mechanics, and, contrary to the quantum set, whose characterization we do not even know to be decidable, it admits a computationally efficient description.
At the time, we conjectured that the almost quantum set corresponds to the set of correlations of a yet-to-be-discovered consistent physical theory; similar to quantum mechanics, albeit arguably more plausible.
Nowadays, we are not so optimistic: in  we show that almost quantum correlations do not emerge from any physical theory satisfying the No-Restriction Hypothesis (NRH). The NRH essentially demands that any mathematically well-defined measurement must admit a physical realization. In a sense, the NRH captures the spirit of the scientific method: we test the validity of scientific theories by making predictions beyond the framework where they are known to be successful. E.g.: from Einstein’s equations we predicted the existence of black holes long before we had any observational evidence. Alternatively, we could have modified the theory of General Relativity to forbid the existence of such unobserved objects.
Despite this blow, there is still hope: the almost quantum set leads naturally to a rich supra-quantum physical theory, with correlations beyond almost quantum . We do not know yet whether his theory satisfies the afore-mentioned physical principles, but there is reason to believe that it does admit a correspondence principle with classical physics. This could lead to testable predictions incompatible with the quantum framework.
 M. Navascués and H. Wunderlich, A glance beyond the quantum model, Proc. Royal Soc. A 466:881-890 (2009).
 B. Lang, T. Vértesi and M. Navascués, Closed sets of correlations: answers from the zoo, Journal of Physics A 47, 424029 (2014).
 M. Navascués, Y. Guryanova, M. J. Hoban, A. Acín, Almost quantum correlations, Nat. Comm. 6, 6288 (2015).
 A. B. Sainz, Y. Guryanova, A. Acín and Miguel Navascués, Almost quantum correlations violate the no-restriction hypothesis, arXiv:1707.02620.
 Work in progress.
N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).