  In this paper, we construct the quantum Torus symmetry of the KP hierarchy
and further derive the quantum torus constraint on the tau function of the KP
hierarchy. That means we give a nice representation of the quantum Torus Lie
algebra in the KP system by acting on its tau function. Comparing to the
$W_{\infty}$ symmetry, this quantum Torus symmetry has a nice algebraic
structure with double indices. Further by reduction, we also construct the
quantum Torus symmetries of the KdV and BKP hierarchies and further derive the
quantum Torus constraints on their tau functions. These quantum Torus
constraints might have applications in the quantum field theory, supersymmetric
gauge theory and so on.
