We introduce an new notation for quantum correction for the Teichmuller space of polarized Calabi-Yau manifolds. Under the assumption of vanishing of weak quantum correction, we show that the Teichmuller spaces, with the Weil-Petersson metric, are locally Hermitian symmetric spaces. For Calabi-Yau threefolds, we prove that vanishing of strong quantum correction is equivalent to that the image of the Teichmuller space under the period map is an open submanifold of a globally Hermitian symmetric space.