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It has long been recognized that certain quantum correlations areincompatible with particular assumption about classical causal structure. Givena causal structure of unknown classicality, the presence of such correlationscertifies the nonclassical nature of the causal structure in adevice-independent fashion. In structures where all parties share a commonresource, these nonclassical correlations are also known as nonlocalcorrelations. Any constraint satisfied by all correlations which areclassically compatible with a given causal structure defines a causalcompatibility criterion. Such criteria were recently derived for the Trianglestructure (E. Wolfe et al., arXiv:1609.00672) in the form of polynomialinequalities, begging the question of whether any of those inequalities admitviolation by quantum correlations. Numerical investigation suggests that theydo not, and we further conjecture that the set of correlations admitted by theclassical Triangle structure is equivalent to the set of correlations admittedby its quantum generalization whenever the three observable variables arebinary. Our main contribution in this work, however, is the derivation of newcausal compatibility inequalities for the Triangle structure which do admitquantum violation. This provides a robust-to-noise witness of quantumcorrelations in the Triangle structure. We conclude by considering thepossibility of quantum resources potentially qualitatively different from thoseknown previously.

Causal Compatibility Inequalities Admitting Quantum Violations in the Triangle Structure