  The absolute/relative debate on the nature of space and time is ongoing for
thousands of years. Here we attempt to investigate space and time from the
information theoretic point of view to understand spatial and temporal
correlations under the relative assumption. Correlations, as a measure of
relationship between two quantities, do not distinguish space and time in
classical probability theory; quantum correlations in space are well-studied
but temporal correlations are not well understood. The thesis investigates
quantum correlations in space-time, by treating temporal correlations equally
in form as spatial correlations and unifying quantum correlations in space and
time. In particular, we follow the pseudo-density matrix formalism in which
quantum states in spacetime are properly defined by correlations from
measurements. We first review classical correlations, quantum correlations in
space and time, to motivate the pseudo-density matrix formalism in finite
dimensions. Next we generalise the pseudo-density matrix formulation to the
Gaussian case, general continuous variables via Wigner representations, and
general measurement processes like weak measurements. Then we compare the
pseudo-density matrix formalism with other spacetime formulations: indefinite
causal structures, consistent histories, generalised non-local games,
out-of-time-order correlation functions, and path integrals. We argue that in
non-relativistic quantum mechanics, different spacetime formulations are
closely related via quantum correlations, except path integrals. Finally, we
apply the pseudo-density matrix formulation to time crystals. By defining time
crystals as long-range order in time, we analyse continuous and discrete time
translation symmetry as well as discuss the existence of time crystals from an
algebraic point of view. Finally, we summarise our work and provide the outlook
for future directions.
