  We first study the quantum product on the big phase space defined by
gravitational Gromov-Witten invariants. We then use this product to give an
interpretation for various topological recursion relations and also use it to
study the Virasoro conjecture proposed by Eguchi-Hori-Xiong and Katz. We will
give a recursive formulation to the Virasoro conjecture and study properties of
relevent vector fields, which will be useful in proving and applying the
Virasoro conjecture for all genera. In the genus-2 case, we will prove that the
genus-2 Virasoro conjecture can be reduced to the $L_{1}$ constraint for any
manifold. In the case when the quantum cohomology of the underlying manifold is
not too degenerate (in particular, is semisimple) we will prove an explicit
formula expressing the generating function of genus-2 Gromov-Witten invariants
in terms of genus-0 and genus-1 data. This result reduces the genus-2 Virasoro
conjecture to a genus-1 problem for such manifolds.
