  In this thesis, we study chiral topological phases of 2+1 dimensional quantum
matter. Such phases are abstractly characterized by their non-vanishing chiral
central charge $c$, a topological invariant which appears as the coefficient of
a gravitational Chern-Simons (gCS) action in bulk, and of corresponding
gravitational anomalies at boundaries. The chiral central charge is of
particular importance in chiral superfluids and superconductors (CSF/Cs), where
$U(1)$ particle-number symmetry is broken, and $c$ is, in some cases, the only
topological invariant characterizing the system. However, as opposed to
invariants which can be probed by gauge fields in place of gravity, the
concrete physical implications of $c$ in the context of condensed matter
physics is quite subtle, and has been the subject of ongoing research and
controversy. The first two parts of this thesis are devoted to the physical
interpretation of the gCS action and gravitational anomalies in the context of
CSF/Cs, where they are of particular importance, but have nevertheless remained
poorly understood. We then turn to a seemingly unrelated aspect of chiral
topological phases - their computational complexity. The infamous $sign\
problem$ leads to an exponential complexity in Monte Carlo simulations of
generic many-body quantum systems. Nevertheless, many phases of matter are
known to admit a sign-problem-free representative, allowing an efficient
classical simulation. The possibility of $intrinsic$ sign problems, where a
phase of matter admits no sign-problem-free representative, was recently raised
but remains largely unexplored. Here, we establish the existence of an
intrinsic sign problem in a broad class of chiral topological phases, both
bosonic and fermionic, defined by the requirement that $e^{2\pi i c/24}$ is
$not$ the topological spin of an anyon.
