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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 zero 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 implies in particular that the real symplectic quotient is graded Gorenstein. In case 
$K$ is a torus or 
$\operatorname{SU}_{2}$, we show that these results hold without the hypothesis that 
$(V,G)$ is 
$3$-large.
