This paper presents a formula for products of Schubert classes in
the quantum cohomology ring of the Grassmannian. We introduce a
generalization of Schur symmetric polynomials for shapes that are
naturally embedded in a torus. Then we show that the coefficients
in the expansion of these toric Schur polynomials, in terms of the
regular Schur polynomials, are exactly the 3-point Gromov-Witten
invariants, which are the structure constants of the quantum
cohomology ring. This construction implies three symmetries of the
Gromov-Witten invariants of the Grassmannian with respect to the
groups $S_3$ , $(\mathbb{Z}/n\mathbb{Z})^2$ , and
$\mathbb{Z}/2\mathbb{Z}$ . The last symmetry is a certain
\emph{curious duality} of the quantum cohomology which inverts the
quantum parameter $q$ . Our construction gives a solution to a
problem posed by Fulton and Woodward about the characterization of
the powers of the quantum parameter $q$ which occur with nonzero
coefficients in the quantum product of two Schubert classes. The
curious duality switches the smallest such power of $q$ with the
highest power. We also discuss the affine nil-Temperley-Lieb
algebra that gives a model for the quantum cohomology.