  We reveal a novel regime of photon-pair generation driven by the interplay of
multiple bound states in the continuum resonances in nonlinear metasurfaces.
This non-degenerate photon-pair generation is derived from the hyperbolic
topology of the transverse phase-matching and can enable orders-of-magnitude
enhancement of the photon rate and spectral brightness, as compared to the
degenerate regime. We show that the entanglement of the photon-pairs can be
tuned by varying the pump polarization, which can underpin future advances and
applications of ultra-compact quantum light sources.
