  It has long been recognized that certain quantum correlations are
incompatible with particular assumption about classical causal structure. Given
a causal structure of unknown classicality, the presence of such correlations
certifies the nonclassical nature of the causal structure in a
device-independent fashion. In structures where all parties share a common
resource, these nonclassical correlations are also known as nonlocal
correlations. Any constraint satisfied by all correlations which are
classically compatible with a given causal structure defines a causal
compatibility criterion. Such criteria were recently derived for the Triangle
structure (E. Wolfe et al., arXiv:1609.00672) in the form of polynomial
inequalities, begging the question of whether any of those inequalities admit
violation by quantum correlations. Numerical investigation suggests that they
do not, and we further conjecture that the set of correlations admitted by the
classical Triangle structure is equivalent to the set of correlations admitted
by its quantum generalization whenever the three observable variables are
binary. Our main contribution in this work, however, is the derivation of new
causal compatibility inequalities for the Triangle structure which do admit
quantum violation. This provides a robust-to-noise witness of quantum
correlations in the Triangle structure. We conclude by considering the
possibility of quantum resources potentially qualitatively different from those
known previously.
