The AdS/CFT correspondence, as a manifestation of the holographic principle, has provided valuable insights while raising more intriguing questions in our pursuit of quantum gravity. In this dissertation, we explore the interconnections between quantum information and quantum chaos in holography, through the lens of entanglement entropy and black holes. We present a series of studies, each contributing a novel perspective or improvement to our understanding of quantum entanglement and gravitational systems.

We begin by introducing a new way for extracting the von Neumann entropy from integer $n$ R enyi entropies using a generating function. This method does not rely on direct analytic continuation in $n$, and we demonstrate its utility through analytical and numerical examples. With the generating function, we establish the expressivity of von Neumann and R enyi entropies in terms of classical and quantum neural networks, showcasing the potential of machine learning in addressing complex problems in quantum information theory.

Furthermore, we utilize the Euclidean gravitational path integral to investigate holographic entanglement entropy under bulk renormalization group flow. This study addresses the consistency of the holographic dictionary, ensuring that different bulk descriptions yield the same boundary entanglement entropy, given its fine-grained nature. We generalize the conical expansion method to examine scenarios involving non-zero spin matter fields in tree-level UV extensions, demonstrating that the UV entropy values concur with those flowed to the IR upon going on-shell. Moreover, we find that the entropy functional consistently aligns under imposition of theequations of motion for the matter fields at low energies, surpassing mere entropy value matching.

As an incarnation of holographic entanglement entropy, the bulk extremal surface leads to the subregion duality paradigm. By defining the entanglement wedge associated with a boundary subregion, one can naturally extrapolate entanglement wedge reconstruction to define a notion of operator size in the boundary and determine the butterfly velocity that characterizes the growth of local perturbations from certain extremal surfaces. On the other hand, the study of quantum chaos presents a novel bound to holographic theories through a distinct Lorentzian calculation of the butterfly velocity, determined from a localized shockwave on the horizon of a dual black hole. We demonstrate a general agreement between the two pictures in higher-derivative gravity, revealing deep connections between quantum chaos and entanglement wedge reconstruction and sharply constraining the paradigm.

Entanglement wedge reconstruction offers a beautiful resolution to the black hole information paradox through the concept of quantum extremal islands. We delve into the dynamics of black hole evaporation within a less understood non-minimal dilaton gravity framework. By identifying the Weyl-invariant terms in the action, which could be attributed to a state-dependent part of the stress tensor, we constructed a one-parameter family of one-loop actions with unique, regular, and physical stress tensors corresponding to quantum states that describe black hole evaporation. We apply the quantum extremal islands prescription to the back-reacted geometry, successfully reproducing the correct Page curve and thereby affirming the unitarity of the evaporation process, which was unattainable without a consistent one-loop theory.