  In this paper, we studied the space-time estimates for the solution to the
Schr\"odinger equation. By polynomial partitioning, induction arguments,
bilinear to linear arguments and broad norm estimates, we set up several
maximal estimates for the Schr\"odinger equation with high-frequency input
data. By these maximal estimates, we obtain the sharp global space-time
estimate when $n=2$ and improve the known results in the critical cases when
$n\geq 3$. The maximal estimate for $n=2$ is also used to extend the results of
the local space-time estimates for the solution to the Schr\"odinger equation.
