By adapting the white noise theory, the quantum analogues of the (classical)
Gross Laplacian and L´evy Laplacian, so called the quantum Gross Laplacian and
quantum L´evy Laplacian, respectively, are introduced as the Laplacians acting
on the spaces of generalized operators. Then the integral representations of the
quantum Laplacians in terms of quantum white noise derivatives are studied. Correspondences
of the classical Laplacians and quantum Laplacians are studied. The
solutions of heat equations associated with the quantum Laplacians are obtained
from a normal-ordered white noise differential equation
