  We first present a protocol for deterministically distilling non-locality,
building upon a recent result of Forster et al. [Phys. Rev. Lett. 102, 120401
(2009)]. Our protocol, which is optimal for two-copy distillation, works
efficiently for a specific class of post-quantum non-local boxes, which we term
correlated non-local boxes. In the asymptotic limit, all correlated non-local
boxes are distilled to the maximally non-local box of Popescu and Rohrlich.
Then, taking advantage of a result of Brassard \textit{et al.} [Phys. Rev.
Lett. 96, 250401 (2006)] we show that all correlated non-local boxes make
communication complexity trivial, and therefore appear very unlikely to exist
in nature. Astonishingly, some of these non-local boxes are arbitrarily close
to the set of classical correlations. This result therefore gives new insight
to the problem of why quantum non-locality is limited.
