Computable properties of quantum states are given a dual gravitational interpretation via the AdS/CFT correspondence. For holographic states, boundary entanglement entropy is dual to the area of bulk geodesics, known as Ryu-Takayanagi surfaces. Furthermore, the viability of states to admit a holographic dual at all is constrained by their entanglement structure. Entanglement therefore defines a coarse classification of states in the Hilbert space. Similarly, how a state transforms under a group of operators also provides a classification on the Hilbert space. Certain states, e.g. stabilizer states, are invariant under large sets of operations, and consequently can be simulated on a classical computer. Cayley graphs offer a useful representation for a group of operators, where vertices represent group elements and edges represent group generators. In this representation, the orbit of a state under action of the group can also be represented as a ``reachability graph'', defined as a quotient of the group Cayley graph. Reachability graphs can be dressed to encode entanglement information, making them a useful tool for studying entanglement dynamics under quantum operations. Further quotienting a reachability graph by group elements that fix a chosen state property, e.g. entanglement entropy, builds a ``contracted graph''. Contracted graphs provide explicit bounds on state parameter evolution under quantum circuits. In this work, an upper bound on entropy vector evolution under Clifford group action is presented. Another important property of quantum systems is magic, which quantifies the difficulty of classically simulating a quantum state. Magic and entanglement are intimately related, but the two are not equivalent measures of complexity. Nonetheless, entanglement and magic play complementary roles when describing emergent gravitational phenomena in AdS/CFT. This manuscript describes the interplay between entanglement and magic, and offers a holographic interpretation for magic as cosmic brane back-reaction.

Much attention has been given to the behavior of quantum fields in expanding Freidmann-Lema\^itre-Robertson-Walker (FLRW) spacetimes, and de Sitter spacetime in particular. In such spacetimes, the S-matrix is ill-defined, so new observables must be constructed that are accessible to both computation and measurement. The most common observable in theories of inflation is an equal-time correlation function, typically computed in the in-in formalism. Weinberg improved upon in-in perturbation theory by reducing the perturbative expansion to a series of nested commutators. Several authors noted a technical difference between Weinberg's formula and standard in-in perturbation theory. In this work, a proof of the order-by-order equivalence of Weinberg's commutators to traditional in-in perturbation theory is presented for all masses and commonly studied spins in a broad class of FLRW spacetimes. Then, a study of the effects of a sector of conformal matter coupled solely to gravity is given. The results can constrain N-naturalness as a complete solution of the hierarchy problem, given a measurement of the tensor fluctuations from inflation. The next part of this work focuses on the thermodynamics of de Sitter. It has been known for decades that there is a temperature associated with a cosmological horizon, which matches the thermal response of a comoving particle detector in de Sitter. A model of a perfectly reflecting cavity is constructed with fixed physical size in two-dimensional de Sitter spacetime. The natural ground state inside the box yields no response from a comoving particle detector, implying that the box screens out the thermal effects of the de Sitter horizon. The total energy inside the box is also shown to be smaller than an equivalent volume of the Bunch-Davies vacuum state. The temperature difference across the wall of the box might drive a heat engine, so an analytical model of the Szil\'ard engine is constructed and studied. It is found that all relevant thermodynamical quantities can be computed exactly at all stages of the engine cycle.

In this dissertation I discuss about calculating one-loop partition function on curved spacetimes and various approaches to build symmetries of gravitational systems, and extending the analysis to the large dimensional spacetimes. I show the calculations pertaining to the contributions to the one-loop determinant for transverse trace-less gravitons in an $n + 3$-dimensional Schwarzschild black hole background in the large dimension limit, due to the $SO(n+2)$-type tensor and vector fluctuations, using the quasinormal mode method. Accordingly I find the quasinormal modes for these fluctuations as a function of a fiducial mass parameter $\Delta$. I show that the behavior of the one-loop determinant at large $\Delta$ accords with a heat kernel curvature expansion in one lower dimension, lending further evidence towards a membrane picture for black holes in the large dimension limit. I also find that the analysis of building one-loop determinants is similar to that of the AdS, thus serving as a motivation to explore this emergent symmetry in detail. For this, I first build these symmetries for Kerr-(A)dS black holes in arbitrary dimensions and then extend this analysis to the large dimensional Schwarzschild black hole. To study the former, in this dissertation, I discuss how to generalize the notion of hidden conformal symmetry in Kerr/CFT to Kerr-(A)dS black holes in arbitrary dimensions. I also discuss the results on building the $SL(2, R)$ generators directly from the Killing tower, whose Killing tensors and Killing vectors enforce the separability of the equations of motion. This construction amounts to an explicit relationship between hidden conformal symmetries and Killing tensors: I use the Killing tower to build a novel tensor equation connecting the $SL(2,R)$ Casimir with the radial Klein-Gordon operator. For asymptotically flat black holes in four and five dimensions I discuss that the previously known results that were obtained using the ``near-region'' limit and the monodromy method, were recovered. I also perform a monodromy evaluation of the Klein-Gordon scalar wave equation for all Kerr-(A)dS black holes, finding explicit forms for the zero mode symmetry generators. Lastly, I discuss the work on extending this analysis to the large-dimensional Schwarzschild black hole as a step towards building a Large-D/CFT correspondence.

Recent developments inspired by string theoretic considerations provide multiple maps between gravitational and non-gravitational degrees of freedom. In this dis- sertation I discuss aspects of three such dualities, the gauge/gravity duality and how it applies to condensed matter systems, the fluid-gravity duality, and the color-kinematics duality.

The first of these, colloquially referred to as holography, in its simplest form posits a mapping of d-dimensional conformal field theory (boundary) partition functions onto d+1 dimensional gravitational(bulk) partition functions, where the space-time carries a negative cosmological constant. In this dissertation I discuss the results of our calculations examining the emergence of Fermi-surface like structures in the bulk spacetime despite the absence of explicit Fermions in the theory.Specifically the 4+1 dimensional Einstein-Maxwell-Chern-Simons theory with scalar degrees of freedom, with and without symmetry breaking is considered. These theories are gravity duals to spatially modulated gauge theories. The results of calculations presented here indicate the existence of a rich phase space, most prominently Fermi shells are seen.

The second set of dualities considered are the color-kinematic duality, also known as the double-copy paradigm and the fluid-gravity duality. The color-kinematic duality involves identifying spin-2 amplitudes as squares of spin-1 gauge amplitudes. This double copy picture is utilized to construct “single copy” representations for space- times where Einstein’s equations reduce to incompressible Navier-Stokes equations. In this dissertation I show how spacetimes that characterize irrotational fluids and constant vorticity fluids each map to distinct algebraically special spacetimes. The Maxwell fields obtained via the double-copy picture for such spacetimes further provide interesting parallels, for instance, the vorticity of the fluid is proportional to the magnetic field of the associated gauge field.

In this dissertation, I present the results from my recent

investigations into the interactions involving topological defects, such as

magnetic monopoles and strings, that may have been produced in the early

universe. I performed numerical studies on the interactions of twisted

monopole-antimonopole pairs in the 't Hooft-Polyakov model for a range of

values of the scalar to vector mass ratio. Sphaleron solution predicted by

Taubes was recovered, and I mapped out its energy and size as functions of

parameters. I also looked into the production, and decay modes of $U(1)$ gauge

and global strings. I demonstrated that strings can be produced upon evolution

of gauge wavepackets defined within a certain region of parameter space. The

numerical exploration of the decay modes of cosmic string loops led to the

conclusions that string loops emit particle radiation primarily due to kink

collisions, and that their decay time due to these losses is proportional to

$L^p$, where $L$ is the loop length and $p \approx 2$. In contrast, the decay

time due to gravitational radiation scales in proportion to $L$, and I

concluded that particle emission is the primary energy loss mechanism for loops

smaller than a critical length scale, while gravitational losses dominate for

larger loops. In addition, I analyzed the decay of cosmic global string loops

due to radiation of Goldstone bosons and massive scalar ($\chi$) particles.

The length of loops I studied ranges from 200-1000 times the width of the

string core. I found that the lifetime of a loop is approximately $1.4L$. The

energy spectrum of Goldstone boson radiation has a $k^{-1}$ fall off, where $k$

is the wavenumber, and a sharp peak at $k\approx m_\chi/2$, where $m_\chi$ is

the mass of $\chi$. The latter is a new feature and implies a peak at high

energies (MeV-GeV) in the cosmological distribution of QCD axions.