Quantum cohomology of complete intersections

The quantum cohomology algebra of a projective manifold $X$ is the cohomology $H^*(X,\mathbf Q)$ endowed with a different algebra structure, which takes into account the geometry of rational curves in $X$. We show that this algebra takes a remarkably simple form for complete intersections when the dimension is large enough with respect to the degree. As a consequence we get a number of enumerative formulas relating lines, conies and twisted cubics on $X$.