  Causal inequalities are device-independent constraints on correlations
realizable via local operations under the assumption of definite causal order
between these operations. While causal inequalities in the bipartite scenario
require nonclassical resources within the process-matrix framework for their
violation, there exist tripartite causal inequalities that admit violations
with classical resources. The tripartite case puts into question the status of
a causal inequality violation as a witness of nonclassicality, i.e., there is
no a priori reason to believe that quantum effects are in general necessary for
a causal inequality violation. Here we propose a notion of classicality for
correlations--termed deterministic consistency--that goes beyond causal
inequalities. We refer to the failure of deterministic consistency for a
correlation as its antinomicity, which serves as our notion of nonclassicality.
Deterministic consistency is motivated by a careful consideration of the
appropriate generalization of Bell inequalities--which serve as witnesses of
nonclassicality for non-signalling correlations--to the case of correlations
without any non-signalling constraints. This naturally leads us to the
classical deterministic limit of the process matrix framework as the
appropriate analogue of a local hidden variable model. We then define a
hierarchy of sets of correlations--from the classical to the most
nonclassical--and prove strict inclusions between them. We also propose a
measure for the antinomicity of correlations--termed 'robustness of
antinomy'--and apply our framework in bipartite and tripartite scenarios. A key
contribution of this work is an explicit nonclassicality witness that goes
beyond causal inequalities, inspired by a modification of the Guess Your
Neighbour's Input (GYNI) game that we term the Guess Your Neighbour's Input or
NOT (GYNIN) game.

Quantum Physics