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Topology has emerged over the years as one of the most powerful theoretical tools to interpret physical phenomena. While its applications to physics have a long history, topological insulators and superconductors have been at the forefront of more recent research activity. The main reason is that topologically-protected edge states and zero-energy Majorana modes arising in topological insulators and superconductors, respectively, appear as ideal candidates for the transfer and storage of energy or information on the one hand, and quantum computation robust against decoherence, on the other hand. Topological insulators have already been investigated in a variety of experimental platforms, but the theoretical understanding of the effects interaction in these systems is still at an early stage. For topological superconductors and superfluids, instead, it is even their physical realization to be essentially still lacking. This is because such states of matter require fermionic particles to pair up into non-zero angular momentum states, primarily in the p-wave channel. Unfortunately, materials that naturally exhibit such a pairing are scarce in nature. With QuDiT, we plan to construct a versatile novel experimental platform that will grant access to both topological p-wave superfluids and to unexplored interacting topological phases on lattice geometries, where the combination of non-local interactions and topology could be analyzed in a controlled way. The novel platform will consist of a Bose-Fermi mixture of magnetic atoms. The highly non-local character of dipole-dipole interactions, the full tunability of additional short range Bose-Fermi and Bose-Bose interactions, as well as the trapping geometry, will allow us to engineer topological fermionic pairing and to access strongly correlated topological regimes. One experimental and two theoretical units will join their diverse and complementary expertise in this effort. The experimental activity will be mainly devoted to the construction of the novel apparatus, while theory will explore in detail the possibilities to produce and control topological phases, paving the way towards their experimental achievement and providing novel predictions. The successful accomplishment of QuDiT will produce a long-lasting advance in our understanding of topological phases, and in their application to novel quantum technologies, with both short- and long-term impacts. In addition, QuDiT will establish a strong synergy between three competitive research groups working at the frontiers of ultracold gases, superfluid and topological matter, reinforcing the national scientific community, and making the three units even more competitive for the participation to the Research and Innovation programs of the European Commission