  We prove that if X is a Grassmannian of type A, then the Schubert basis of
the (small) quantum cohomology ring QH(X) is the only homogeneous deformation
of the Schubert basis of the ordinary cohomology ring of X that multiplies with
non-negative structure constants. This implies that the (three point, genus
zero) Gromov-Witten invariants of X are uniquely determined by Witten's
presentation of QH(X) and the fact that they are non-negative. We conjecture
that the same is true for any flag variety X = G/P of simply laced Lie type.
For the variety GL(n)/B of complete flags, this conjecture is equivalent to
Fomin, Gelfand, and Postnikov's conjecture that the quantum Schubert
polynomials of type A are uniquely determined by positivity properties. Our
proof for Grassmannians answers a question of Fulton.
