Abstract: We present recent advances in our understanding of (i) exotic quantum phases of matter in three dimensions, and (ii) robust mechanisms for storing and processing quantum information, that have been enabled by new techniques to study highly-entangled quantum states. First, we introduce new kinds of gapped quantum phases in three spatial dimensions, which are characterized by the presence of point-like excitations that are strictly immobile at zero temperature, and by degenerate ground-states that are locally indistinguishable. These states of matter -- termed "fracton" phases -- provide an intriguing alternative to the conventional Bose or Fermi statistics of point-like excitations in three dimensions, as well as a gateway for studying "slow" dynamical behavior in the absence of disorder.
After presenting systematic constructions of fracton phases, we apply similar techniques in order to construct entangled states of fermions that can be used to robustly store and process quantum information; our efforts are inspired by recent interest in using the fermion occupation number to encode physical qubits. We introduce the first quantum error-correcting codes that can recover from both fermion parity-violating (quasiparticle poisoning) and parity-preserving errors in Fermi systems, and propose physical realizations of these codes in on-going experiments on mesoscopic superconductors that host Majorana zero modes.