A powerful tool to study Calabi-Yau manifolds is the two-dimensional N=(2,2) gauged linear sigma model. Using the two-sphere partition function of such two-dimensional gauge theories, I present a new approach to calculate the exact Kähler potential of the quantum Kähler moduli space of Calabi-Yau manifolds. In particular this allows one to compute the genus zero Gromov-Witten invariants for Calabi-Yau manifolds without the use of mirror symmetry. In this way, the gauged linear sigma model with abelian gauge groups reproduces known Gromov-Witten invariants for Calabi-Yau hypersurfaces in toric varieties. For gauged linear sigma models with non-abelian gauge groups we obtain new predictions of Gromov-Witten invariants for a more general class of geometries such as determinantal Calabi-Yau threefolds.