  Dynamical observables can often be described by time correlation functions
(TCFs). However, efficiently calculating TCFs for complex quantum systems is a
significant challenge, which generally requires solving the full dynamics of
the systems. This Letter presents the memory kernel coupling theory (MKCT), a
general formalism for evaluating TCFs. The MKCT builds upon Mori's memory
kernel formalism for TCFs. Our theory further decomposes the memory kernel into
auxiliary kernels. Rapid decay of auxiliary kernels allows us to truncate the
coupled equations of motion with high accuracy. Notably, only higher-order
moments are sufficient as the input for obtaining TCFs. While this formalism is
general, we carry out the numerical demonstration for a typical open quantum
system--the spin-boson model.
