Abstract: In this talk, we will construct a family of complete bases for the non-perturbative Hilbert space of quantum gravity and use this to address two puzzles regarding the Hilbert space of quantum gravity. Firstly, Gibbons and Hawking proposed that the Euclidean gravity path integral with periodic boundary conditions in time computes the thermal partition sum of gravity. As a corollary, the derivative of the associated free energy with respect to the Euclidean time period results in the celebrated black hole entropy formula S=A/4G. Why is this interpretation correct? That is, why does this path integral compute a trace over the gravity Hilbert space? We show that the quantity computed by the Gibbons-Hawking path integral is equal to an a priori different object -- an explicit thermal trace over the Hilbert space spanned by states produced by the Euclidean gravity path integral. Secondly, Holography suggests that the Hilbert space of quantum gravity with two asymptotic boundaries should factorise into two copies of the single-boundary gravity Hilbert space.  However, the existence of spacetimes with Einstein-Rosen bridges connecting the two boundaries seems to contradict this. We will show that the non-perturbative, two-boundary Hilbert space factorises nonetheless. One implication of our results is that universes containing a horizon can sometimes be understood as superpositions of horizonless geometries entangled with a closed universe.