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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, worksefficiently for a specific class of post-quantum non-local boxes, which we termcorrelated non-local boxes. In the asymptotic limit, all correlated non-localboxes 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 makecommunication complexity trivial, and therefore appear very unlikely to existin nature. Astonishingly, some of these non-local boxes are arbitrarily closeto the set of classical correlations. This result therefore gives new insightto the problem of why quantum non-locality is limited.

Non-locality distillation and post-quantum theories with trivial communication complexity