Among complex systems with emergent behaviours, frustrated quantum magnets are predicted to exhibit novel, highly nontrivial phases of matter that may play a major role in future and emerging quantum technologies such as the synthesis of innovative materials for energy harnessing and storage, entanglement-enhanced metrology, and topological quantum computing.Unfortunately, due to the intrinsic levels of noise in "natural" compounds, the controlled realization, characterization, and manipulation of frustrated quantum magnets appear exceedingly demanding. On the other hand, we are now entering an advanced stage of development of quantum emulators, engineered quantum systems that realize model Hamiltonians of increasing complexity in a controlled fashion. Cutting-edge technologies for quantum emulation science include cold atoms in optical lattices, trapped ultracold ions (Coulomb crystals), NV centres in diamond, and photonic circuits. By developing, comparing, and integrating these four different atom-optical platforms, project EQuaM's breakthrough is the controlled experimental emulation of fundamental model Hamiltonians for frustrated quantum magnetism, both in nontrivial lattice geometries and for competing long-range interactions, and the characterization of their phase diagrams, targeting fundamental features such as spin liquid phases, global topological order, and fractional excitations. By achieving this objective, EQuaM's groundbreaking contribution to the long-term vision in Information and Communication Technologies (ICT) is the efficient quantum emulation, not admitting efficient classical computational counterparts, of many-body quantum systems with essential elements of complexity. Besides providing crucial insights in the physics of complex many-body systems, it will be a foundational step in the realization of large-scale architectures for topologically protected quantum computation and information.