In the holographic setting of gauge/gravity duality, there has been a growing expectation that the emergence of the bulk gravitational spacetime is rooted in the entanglement structure of the dual gauge theory. While the deep connection remains mysterious, an early hint came from the holographic entanglement entropy prescription, which expresses the boundary entanglement entropy in terms of simple geometric constructs in the bulk. This talk will review the recent reformulations of this prescription presented in [2208.10507] in terms of several new (and fully covariant) formulas, including minimax, max U-flow, and min V-flow (the latter covariantizing the bit thread formulation of holographic entanglement entropy by Freedman and Headrick).