  Nonlocal games with synchronous correlations are a natural generalization of
functions between two finite sets. In this work we examine analogues of Bell's
inequalities for such correlations, and derive a synchronous device-independent
quantum key distribution protocol. This protocol has the advantage of symmetry
between the two users and self-testing while generating shared secret key
without requiring a preshared secret. 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 prove existence and rigidity of quantum correlations
that saturate this bound. We conclude by posing a security assumption that
bypasses the locality, or causality, loophole and examine the protocol's
robustness against measurement error and depolarization noise.
