  We take a resource-theoretic approach to the problem of quantifying
nonclassicality in Bell scenarios. The resources are conceptualized as
probabilistic processes from the setting variables to the outcome variables
having a particular causal structure, namely, one wherein the wings are only
connected by a common cause. We term them "common-cause boxes". We define the
distinction between classical and nonclassical resources in terms of whether or
not a classical causal model can explain the correlations. One can then
quantify the relative nonclassicality of resources by considering their
interconvertibility relative to the set of operations that can be implemented
using a classical common cause (which correspond to local operations and shared
randomness). We prove that the set of free operations forms a polytope, which
in turn allows us to derive an efficient algorithm for deciding whether one
resource can be converted to another. We moreover define two distinct monotones
with simple closed-form expressions in the two-party binary-setting
binary-outcome scenario, and use these to reveal various properties of the
pre-order of resources, including a lower bound on the cardinality of any
complete set of monotones. In particular, we show that the information
contained in the degrees of violation of facet-defining Bell inequalities is
not sufficient for quantifying nonclassicality, even though it is sufficient
for witnessing nonclassicality. Finally, we show that the continuous set of
convexly extremal quantumly realizable correlations are all at the top of the
pre-order of quantumly realizable correlations. In addition to providing new
insights on Bell nonclassicality, our work also sets the stage for quantifying
nonclassicality in more general causal networks.
