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The outcomes of measurements on entangled quantum systems can be nonlocallycorrelated. However, while it is easy to write down toy theories allowingarbitrary nonlocal correlations, those allowed in quantum mechanics arelimited. Quantum correlations cannot, for example, violate a principle known asmacroscopic locality, which implies that they cannot violate Tsirelson's bound.This work shows that there is a connection between the strength of nonlocalcorrelations in a physical theory, and the structure of the state spaces ofindividual systems. This is illustrated by a family of models in which localstate spaces are regular polygons, where a natural analogue of a maximallyentangled state of two systems exists. We characterize the nonlocalcorrelations obtainable from such states. The family allows us to study thetransition between classical, quantum, and super-quantum correlations, byvarying only the local state space. We show that the strength of nonlocalcorrelations - in particular whether the maximally entangled state violatesTsirelson's bound or not - depends crucially on a simple geometric property ofthe local state space, known as strong self-duality. This result is seen to bea special case of a general theorem, which states that a broad class ofentangled states in probabilistic theories - including, by extension, allbipartite classical and quantum states - cannot violate macroscopic locality.Finally, our results show that there exist models which are locally almostindistinguishable from quantum mechanics, but can nevertheless generatemaximally nonlocal correlations.

Limits on non-local correlations from the structure of the local state space