  We describe the inequalities on the possible eigenvalues of products of
unitary matrices in terms of quantum Schubert calculus. Related problems are
the existence of flat connections on the punctured two-sphere with prescribed
holonomies, and the decomposition of fusion product of representations of
SU(n), in the large level limit. In the second part of the paper we investigate
how various aspects of the problem (symmetry, factorization) relate to
properties of the Gromov-Witten invariants.
