  This paper is devoted to constructing a quantum version of the famous KP
hierarchy, by deforming its second Hamiltonian structure, namely the nonlinear
$\hat{W}_{\infty}$ algebra. This is achieved by quantizing the conformal
noncompact $SL(2,R)_{k}/U(1)$ coset model, in which $\hat{W}_{\infty}$ appears
as a hidden current algebra. For the quantum $\hat{W}_{\infty}$ algebra at
level $k=1$, we have succeeded in constructing an infinite set of commuting
quantum charges in explicit and closed form. Using them a completely integrable
quantum KP hierarchy is constructed in the Hamiltonian form. A two boson
realization of the quantum $\hat{W}_{\infty}$ currents has played a crucial
role in this exploration.
