  We give a rigorous, quantitative derivation of the incompressible Euler
equation from the many-body problem for $N$ bosons on $\mathbb{T}^d$ with
binary Coulomb interactions in the semiclassical regime. The coupling constant
of the repulsive interaction potential is $~1/(\varepsilon^2 N)$, where
$\varepsilon \ll 1$ and $N\gg 1$, so that by choosing
$\varepsilon=N^{-\lambda}$, for appropriate $\lambda>0$, the scaling is
supercritical with respect to the usual mean-field regime. For approximately
monokinetic initial states with nearly uniform density, we show that the
density of the first marginal converges to 1 as $N\rightarrow\infty$ and
$\hbar\rightarrow 0$, while the current of the first marginal converges to a
solution $u$ of the incompressible Euler equation on an interval for which the
equation admits a classical solution. In dimension 2, the dependence of
$\varepsilon$ on $N$ is essentially optimal, while in dimension 3, heuristic
considerations suggest our scaling is optimal. Our proof is based on a Gronwall
relation for a quantum modulated energy with an appropriate corrector and is
inspired by recent work of Golse and Paul arXiv:1912.06750 on the derivation of
the pressureless Euler-Poisson equation in the classical and mean-field limits
and of Han-Kwan and Iacobelli arXiv:2006.14924 and the author arXiv:2104.11723
on the derivation of the incompressible Euler equation from Newton's second law
in the supercritical mean-field limit. As a byproduct of our analysis, we also
derive the incompressible Euler equation from the Schr\"odinger-Poisson
equation in the limit as $\hbar+\varepsilon\rightarrow 0$, corresponding to a
combined classical and quasineutral limit.
