  We describe the relationship between the GW approximation and various
equation-of-motion (EOM) coupled-cluster (CC) theories. We demonstrate the
exact equivalence of the G$_0$W$_0$ approximation and the propagator theory for
an electron-boson problem in a particular excitation basis. From there, we
establish equivalence within the quasi-boson picture to the IP+EA-EOM unitary
coupled-cluster propagator. We analyze the incomplete description of screening
provided by the standard similarity-transformed IP+EA-EOM-CC and the recently
introduced G$_0$W$_0$ Tamm-Dancoff approximation. We further consider the
approximate decoupling of IP and EA sectors in EOM-CC treatments and devise the
analogous particle-hole decoupling approach for the G$_0$W$_0$ approximation.
Finally, we numerically demonstrate the exact relationships and magnitude of
the approximations in calculations of a set of molecular ionization potentials
and electron affinities.
