Emergent gauge fields and Coulomb liquids have long been central to the physics of frustrated pyrochlore magnets, yet their realization beyond conventional, i.e., rank-one U(1), spin ice and into fully quantum higher-rank regimes has remained elusive. Here we provide strong evidence for this physics in the spin-12 quantum Heisenberg antiferromagnet on the breathing pyrochlore lattice with symmetry-allowed Dzyaloshinskii-Moriya interactions, using the pseudofermion functional renormalization group. We find a robust quantum analog of the rank-one U(1) Coulomb liquid together with compelling signatures of a putative quantum analog of the rank-two U(1) Coulomb liquid, distinguished by their characteristic multifold pinch-point morphologies in momentum space. This Letter therefore establishes a minimal three-dimensional setting in which signatures of gauge theories of different rank emerge within a single microscopic spin Hamiltonian. In addition, quantum fluctuations qualitatively reshape the classical nearest-neighbor atlas of phases, causing an incommensurate spiral instability and an extended nondipolar quantum paramagnetic regime, both absent in the classical model. Our results establish the breathing pyrochlore as a promising and experimentally relevant platform where higher-rank gauge constraints, conventional magnetic order, and fluctuation-driven quantum phases compete on equal footing, opening a direct route to diagnosing emergent gauge structure in three-dimensional quantum magnets.