  We study the nonlinear Schr\"odinger equation (NLS) with the quadratic
nonlinearity $|u|^2$, posed on the two-dimensional torus $\mathbb{T}^2$. While
the relevant $L^3$-Strichartz estimate is known only with a derivative loss, we
prove local well-posedness of the quadratic NLS in $L^2(\mathbb{T}^2)$, thus
resolving an open problem of thirty years since Bourgain (1993). In view of
ill-posedness in negative Sobolev spaces, this result is sharp. We establish a
crucial bilinear estimate by separately studying the non-resonant and nearly
resonant cases. As a corollary, we obtain a tri-linear version of the
$L^3$-Strichartz estimate without any derivative loss.
