All noncontextuality inequalities for the n-cycle scenario

The problem of separating classical from quantum correlations is, in general, intractable and has been solved explicitly only in a few cases. In particular, known methods cannot provide general solutions for an arbitrary number of settings. We provide the complete characterization of the classical correlations and the corresponding maximal quantum violations for the case of n >= 4 observables X-0, ... , Xn-1, where each consecutive pair {X-i, Xi+1}, sum mod n, is jointly measurable. This generalizes both the Clauser-Horne-Shimony-Holt and the Klyachko-Can-Binicioglu-Shumovsky scenarios, which are the simplest ones for locality and noncontextuality, respectively. In addition, we provide explicit quantum states and settings with maximal quantum violation and minimal quantum dimension.