  In this paper, we consider the Cauchy problem in $\mathbb{R}^N$, $N\geq1$,
for semi-linear Schr\"odinger equations with space-time fractional derivatives.
We discuss the nonexistence of global $L^1$ or $L^2$ weak solutions in the
subcritical and critical cases under some conditions on the initial data and
the nonlinear term. Furthermore, the nonexistence of local $L^1$ or $L^2$ weak
solutions in the supercritical case are studied.
