We study the two-dimensional periodic nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity $|u|^2$. In particular, we study the quadratic NLS with random initial data distributed according to a fractional derivative (of order $α\geq 0$) of the Gaussian free field. After removing the singularity at the zeroth frequency, we prove that the quadratic NLS is almost surely locally well-posed for $α< \frac{1}{2}$ and is probabilistically ill-posed for $α\geq \frac{3}{4}$ in a suitable sense. These results show that in the case of rough random initial data and a quadratic nonlinearity, the standard probabilistic well-posedness theory for NLS breaks down before reaching the critical value $α= 1$ predicted by the scaling analysis due to Deng, Nahmod, and Yue (2019), and thus this paper is a continuation of the work by Oh and Okamoto (2021) on stochastic nonlinear wave and heat equations by building an analogue for NLS.