Nonlocal games, synchronous correlations, and Bell inequalities

A nonlocal game with a synchronous correlation is a natural generalization of a function between two finite sets, and has recently appeared in the context of quantum graph homomorphisms. In this work we examine analogues of Bell's inequalities for synchronous correlations. We show that, unlike general correlations and the CHSH inequality, there can be no quantum Bell violation among synchronous correlations with two measurement settings. However we exhibit explicit analogues of Bell's inequalities for synchronous correlations with three measurement settings and two outputs, provide an analogue of Tsirl'son's bound in this setting, and give explicit quantum correlations that saturate this bound.