We study the quadratically regularized optimal transport (QOT) problem for quadratic cost and compactly supported marginals $μ$ and $ν$. It has been empirically observed that the optimal coupling $π_ε$ for the QOT problem has sparse support for small regularization parameter $ε>0.$ In this article we provide the first quantitative description of this phenomenon in general dimension: we derive bounds on the size and on the location of the support of $π_ε$ compared to the Monge coupling. Our analysis is based on pointwise bounds on the density of $π_ε$ together with Minty's trick, which provides a quadratic detachment from the optimal transport duality gap. In the self-transport setting $μ=ν$ we obtain optimal rates of order $ε^{\frac{1}{2+d}}.$