Soon after the discovery of the cuprate high-temperature superconductors, P. W. Anderson suggested in 1987 that their physics is connected to insulators in which the spins of stationary electrons are highly quantum-entangled. Much theoretical progress has been made in understanding such insulating states with many-boson (i.e. spin) entanglement since then.

However, superconductivity requires mobile fermionic electrons, and I will describe the theory of two classes of quantum states with many-mobile-fermion entanglement. 

(A) the Fractionalized Fermi liquid (FL*), which was predicted to have properties consistent with recent angle-dependent magnetoresistance (ADMR) experiments on the under-hole-doped cuprate superconductors;

(B) the Sachdev-Ye-Kitaev model, and its impact on the theory of the strange metal phase of the cuprates. I will also briefly note the impact of the SYK model on the theory of charged black holes in asymptotically flat 3+1 dimensional space.