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This paper is devoted to constructing a quantum version of the famous KPhierarchy, by deforming its second Hamiltonian structure, namely the nonlinear$\hat{W}_{\infty}$ algebra. This is achieved by quantizing the conformalnoncompact $SL(2,R)_{k}/U(1)$ coset model, in which $\hat{W}_{\infty}$ appearsas a hidden current algebra. For the quantum $\hat{W}_{\infty}$ algebra atlevel $k=1$, we have succeeded in constructing an infinite set of commutingquantum charges in explicit and closed form. Using them a completely integrablequantum KP hierarchy is constructed in the Hamiltonian form. A two bosonrealization of the quantum $\hat{W}_{\infty}$ currents has played a crucialrole in this exploration.Comment: 33

On the Kadomtsev–Petviashvili hierarchy, Ŵ∞ algebra, and conformal SL(2,R)/U(1) model. II. The quantum case