"The physics of complex quantum systems with controllable interactions is emerging as a fundamental topic for a broad community, providing an opportunity to test theories of strongly correlated quantum many-body systems and opening interesting applications such as quantum simulators. Recently, in solid-state structures with effective photon-photon interactions the rich physics of quantum fluids of light has been explored, albeit not yet in the regime of strong photonic correlations. Exciting advances in cavity Quantum Electro-Dynamics (QED) and superconducting circuit QED make strong photon-photon interactions now accessible. A growing interest is focusing on lattices of coupled resonators, implementing Hubbard-like Hamiltonians for photons injected by pump driving fields. Similarly to electronic systems, the physics of large two-dimensional (2D) photonic lattices is a fundamental theoretical challenge in the regime of strong correlations. CORPHO has the ambition to develop novel scalable theoretical methods for 2D lattices of cavities, including spatially inhomogeneous driving and dissipation. The proposed methods are based on a hybrid strategy combining cluster mean-field theory and Wave Function Monte Carlo on a physical ‘Corner’ of the Hilbert space in order to calculate the steady-state density matrix and the properties of the non-equilibrium phases. We will study 2D lattices with complex unit cells and ‘fractional’ driving (only a fraction of the sites is pumped), a configuration that, according to recent preliminary studies, is expected to dramatically enhance and enrich quantum correlations. We will also investigate the interplay between driving and geometric frustration in 2D lattices with polarization-dependent interactions. Finally, the quantum control of strongly correlated photonic systems will be explored, including quantum feedback processes, cooling of thermal fluctuations and switching between multi-stable phases."