We identify significant quantum many-body effects, robust to position fluctuations and strong dipole--dipole interactions, in the forward light scattering from planar arrays and uniform-density disks of cold atoms, by comparing stochastic electrodynamics simulations of a quantum master equation and of a semiclassical model that neglects quantum fluctuations. Quantum effects are pronounced at high atomic densities, light close to saturation intensity, and around subradiant resonances. We show that such conditions also maximize spin--spin correlations and entanglement of formation for the atoms, revealing the microscopic origin of light-induced quantum effects. In several regimes of interest, an enhanced semiclassical model with a single-atom quantum description reproduces light transmission remarkably well, and permits analysis of otherwise numerically inaccessible large ensembles, in which we observe collective many-body analogues of resonance power broadening, vacuum Rabi splitting, and significant suppression in cooperative reflection from atomic arrays.
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