  In this paper we prove the uniqueness and stability in determining a
time-dependent nonlinear coefficient $\beta(t, x)$ in the Schr\"odinger
equation $(i\partial_t + \Delta + q(t, x))u + \beta u^2 = 0$, from the boundary
Dirichlet-to-Neumann (DN) map. In particular, we are interested in the partial
data problem, in which the DN-map is measured on a proper subset of the
boundary. We show two results: a local uniqueness of the coefficient at the
points where certain type of geometric optics (GO) solutions can reach; and a
stability estimate based on the unique continuation property for the linear
equation.
