We study the high-dimensional asymptotics of empirical risk minimization (ERM) in over-parametrized two-layer neural networks with quadratic activations trained on synthetic data. We derive sharp asymptotics for both training and test errors by mapping the $\ell_2$-regularized learning problem to a convex matrix sensing task with nuclear norm penalization. This reveals that capacity control in such networks emerges from a low-rank structure in the learned feature maps. Our results characterize the global minima of the loss and yield precise generalization thresholds, showing how the width of the target function governs learnability. This analysis bridges and extends ideas from spin-glass methods, matrix factorization, and convex optimization and emphasizes the deep link between low-rank matrix sensing and learning in quadratic neural networks.