We show that the Gromov-Witten theory of Calabi- Yau hypersurfaces matches, in genus zero and after an analytic continuation, the quantum singularity theory (FJRW theory) recently introduced by Fan, Jarvis and Ruan following a proposal of Witten. Moreover, on both sides, we highlight two remarkable integral local systems arising from the common formalism of (Gamma) over cap -integral structures applied to the derived category of the hypersurface {W = 0} and to the category of graded matrix factorizations of W. In this setup, we prove that the analytic continuation matches Orlov equivalence between the two above categories.