Abstract
A "quantum-first" approach to gravity is described, where rather than quantizing general relativity, one seeks to formulate the physics of gravity within a quantum-mechanical framework with suitably general postulates. Important guides are the need for appropriate mathematical structure on Hilbert space, and correspondence with general relativity and quantum field theory in weak-gravity situations. A basic physical question is that of "Einstein separability:" how to define mutually independent subsystems, e.g. through localization. Standard answers via tensor products or operator algebras conflict with properties of gravity, as is seen in the correspondence limit; this connects with discussions of "soft hair." Instead, gravitational behavior suggests a networked Hilbert space structure. This structure plus unitarity provide important clues towards a quantum formulation of gravity.