  Via nonlocality distillation, a number of copies of a given nonlocal
correlation can be turned into a new correlation displaying a higher degree of
nonlocality. Apart from its clear relevance in situations where nonlocality is
a resource, distillation protocols also play an important role in the
understanding of information-theoretical principles for quantum theory. Here,
we derive a necessary condition for nonlocality distillation from two copies
and apply it, among other results, to show that $1$D and $2$D quantum voids
--faces of the nonlocal simplex set with no quantum realization-- can be
distilled up to PR-boxes. With that, we generalize previous results in the
literature. For instance, showing a broad class of post-quantum correlations
that make communication complexity trivial and violate the information
causality principle.
