Many-body entanglement lies at the heart of the current developement of quantum technologies. While the realization of large-scale entanglement remains an outstanding challenge, we propose within HighDy a new approach based on the manipulation of highly magnetic dysprosium atoms. We will first aim at generating entanglement among atomic spins in a single-mode Bose-Einstein condensate. We propose a new route based on the initial preparation of each atomic spin in a highly non-classical state. Interactions between spins then induce a collective spin magnification, leading to Dicke squeezing with entanglement-assisted metrological capabilities. Given the complexity of interactions between highly magnetic atoms, a joint experimental and theoretical effort will be required to design the optimal entangling protocol. Our second objective will be the study of Bose-Einstein condensates in a quantum Hall structure. An effective magnetic field will be produced via the light-induced coupling between the atomic motion and the spin, which acts as a synthetic dimension. By controlling the interaction range, we will create Bose-Einstein condensates with a regular lattice of quantized vortices. We will study quantum fluctuations of the vortex lattice, and generate quantum correlations between atoms by quenching the system to a flat-band regime. Lastly, we will integrate ideas from both objectives to develop a novel type of inertial quantum-enhanced sensor. This sensor will be based on spin/momentum locking resulting from light-induced spin-orbit coupling.