Universal scheme for violation of local realism from quantum advantage in one-way communication complexity

We consider relations between communication complexity problems and detecting correlations (violating local realism) with no local hidden variable model. We show first universal equivalence between characteristics of protocols used in that type of problems and non-signaling correlations. We construct non linear bipartite Bell type inequalities and strong nonlocality test with binary observables by providing general method of Bell inequalities construction and showing that existence of gap between quantum and classical complexity leads to violation of these inequalities. We obtain, first to our knowledge, explicit Bell inequality with binary observables and exponential violation.