  The theory of quantum chromodynamics (QCD) encodes the strong interactions
that bind quarks and gluons into nucleons and that bind nucleons into nuclei.
Predictive control of QCD would allow nuclear structure and reactions as well
as properties of supernovae and neutron stars to be theoretically studied from
first principles. Lattice QCD can represent generic QCD predictions in terms of
well-defined path integrals, but the sign and signal-to-noise problems have
obstructed lattice QCD calculations of large nuclei and nuclear matter in
practice. This thesis presents a statistical study of lattice QCD correlation
functions, with a particular focus on characterizing the structure of the noise
associated with quantum fluctuations. The signal-to-noise problem in baryon
correlation functions is demonstrated to arise from a sign problem associated
with Monte Carlo sampling of complex correlation functions. The phases of
complex correlation functions are analyzed in the framework of circular
statistics, and the time evolution of the phase is shown to resemble a
heavy-tailed random walk on the unit circle. Building on these observations, a
new technique called phase reweighting is investigated that involves
calculations of phase differences over fixed-length time intervals. Phase
reweighting allows results for hadronic observables to be extracted from
large-time correlation functions with constant signal-to-noise ratios. The
signal-to-noise problem re-emerges as the length of the phase-difference
interval is increased. Applications of phase reweighting to meson, baryon, and
two-baryon systems are discussed.
