Let$G$be a simple and simply-connected complex algebraic group,$P$⊂$G$a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology$QH$^{*}($G$/$P$) of a flag variety is, up to localization, a quotient of the homology$H$_{*}(Gr_{$G$}) of the affine Grassmannian Gr_{$G$}of$G$. As a consequence, all three-point genus-zero Gromov–Witten invariants of$G$/$P$are identified with homology Schubert structure constants of$H$_{*}(Gr_{$G$}), establishing the equivalence of the quantum and homology affine Schubert calculi.