  Simulating quantum nonlocality and steering requires augmenting pre-shared
randomness with non-vanishing communication cost. This prompts the question of
how one may provide such an operational characterization for the quantumness of
correlations due even to unentangled states. Here we show that for a certain
class of states, such quantumness can be pointed out by superlocality, the
requirement for a larger dimension of the pre-shared randomness to simulate the
correlations than that of the quantum state that generates them. This provides
an approach to define the nonclassicality of local multipartite correlations in
convex operational theories.
