Using a version of Witten´s Dictionary, relating conformal blocks and the quantum cohomology of Grassmannians, in joint work with Prakash Belkale and Swarnava Mukhopadhyay, we show that above the critical level, which we introduce, all vector bundles of type A conformal blocks on \overline M_{0,n} are trivial. We uncover new level-rank symmetries between pairs of critical level conformal blocks divisors and other identities as well. In this talk I will describe the critical level symmetries and some of our applications and put them into perspective in terms of the guiding conjectures about the birational geometry of the moduli space.