  Let $K$ be a compact Lie group with complexification $G$, and let $V$ be a
unitary $K$-module. We consider the real symplectic quotient $M_0$ at level $0$
of the homogeneous quadratic moment map as well as the complex symplectic
quotient, defined here as the complexification of $M_0$. We show that if $(V,
G)$ is $3$-large, a condition that holds generically, then the complex
symplectic quotient has symplectic singularities and is graded Gorenstein. This
in particular implies that the real symplectic quotient is graded Gorenstein.
In the case that $K$ is a torus or $\operatorname{SU}_2$, we show that these
results hold without the hypothesis that $(V,G)$ is $3$-large.
