  We show how nonclassical correlations in local bipartite states can act as a
resource for quantum information processing. Considering the task of quantum
random access codes (RAC) through separable Bell-diagonal states, we
demonstrate the advantage of superunsteerability over classical protocols
assisted with two-bits of shared randomness. We propose a measure of
superunsteerability, which quantifies nonclassicality beyond quantum steering,
and obtain its analytical expression for Bell-diagonal states in the context of
the two- and three-setting steering scenarios that are directly related to the
quantum $2 \to 1$ and $3 \to 1$ RAC protocols, respectively. The maximal values
of our quantifier yield the optimal quantum efficiency for both of the above
protocols, thus showing that superunsteerability provides a precise
characterization of the nonclassical resource for implementing RACs with
separable Bell-diagonal class of states.
