This thesis elucidates various aspects of spacetime emergence from correlations among microscopic degrees of freedom. Holographic dualities such as the AdS/CFT correspondence provide a concrete realization of this idea, and serve as an invaluable testing ground for advancing our understanding of quantum gravity.

Part I examines the entanglement structure of holographic states. Evidence suggest that bipartite pure state entanglement alone is insufficient to account for bulk connectedness. Thus, for some simple holographic states, we consider two mixed-state separability criteria: the positive-partial-transpose (PPT) and the computable cross-norm or realignment (CCNR). In Chapter 2, we take a first step towards characterizing the detailed entanglement structure of Hawking radiation in toy models of black hole evaporation. We apply the PPT criterion in the form of entanglement negativity to detect the onset of entanglement between subsystems of the radiation, finding spacetime wormholes which break the replica symmetry spontaneously. In Chapter 3, we initiate the study of the CCNR criterion in holographic duality. Based on computations in random tensor networks and two-dimensional holographic CFTs, we propose a holographic dual for general semiclassical states. We note the occurrence of a connected-to-disconnected phase transition, which in certain cases can be interpreted as a Hawking-Page transition.

Part II is devoted to the holographic dictionary, with a focus on conjectured entries relating entanglement and chaos to the emergence of bulk locality. We perform explicit checks of these conjectures within the framework of gravitational effective field theories (EFTs), which furnish a controlled laboratory in which to test prospective general properties of holographic duality. One such property is the invariance of boundary observables under bulk renormalization group (RG) flow. In Chapter 4, we study the behavior of holographic entanglement entropy under bulk RG flow in simple examples of gravity coupled to massive fields (which flow to theories with higher-curvature couplings) and find it is indeed invariant. We argue this agreement persists for general tree-level RG flow. In Chapter 5, we consider two holographic definitions of operator size and the butterfly velocity, one based on out-of-time-order correlators (OTOC), dual to certain shockwave spacetimes, and another using entanglement wedge reconstruction. By direct computation, we prove the two definitions of butterfly velocity are equal in a large class of bulk theories with higher-curvature couplings. We also present evidence this equivalence holds in all gravitational theories. The matching provides a highly nontrivial check of operator reconstruction in the entanglement wedge as a general property of holographic duality.

Finally, Part III explores a new twist on holographic duality, in which the bulk theory is dual to an ensemble of boundary theories. We focus on two-dimensional quantum gravity, where the equivalence between the two sides can be made mathematically precise. A canonical example is the duality between Jackiw-Teitelboim (JT) gravity, a two-dimensional dilaton gravity, and a statistical ensemble of random matrices. In Chapter 6, we extend this ensemble duality to a large class of dilaton gravity theories which can be viewed as deforming JT gravity by a gas of conical defects. A novel aspect of these theories is the dynamical merging of conical defects, prohibiting standard approaches for constructing the gravitational path integral. To address this, we exploit a relation between a certain noncritical string theory, deformed by marginal operators, and JT gravity with conical defects, thereby explicitly constructing its random matrix dual. Based on the result, we conjecture a dual random matrix ensemble for general two-dimensional dilaton gravity theories.