The dislocation-mediated quantum melting of solids into quantum liquid crystals is extended from two to three spatial dimensions, using a generalization of boson-vortex or Abelian-Higgs duality. Dislocations are now Burgers-vector-valued strings that trace out worldsheets in spacetime while the phonons of the solid dualize into two-form (Kalb-Ramond) gauge fields. We propose an effective dual Higgs potential that allows for restoring translational symmetry in either one, two or three directions, leading to the quantum analogues of columnar, smectic or nematic liquid crystals. In these phases, transverse phonons turn into gapped, propagating modes while compressional stress remains massless. Rotational Goldstone modes emerge whenever translational symmetry is restored. We also consider electrically charged matter, and find amongst others that as a hard principle only two out of the possible three rotational Goldstone modes are observable using electromagnetic means.
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