  Here, we build on the works of Scuseria (et al.)
http://dx.doi.org/10.1063/1.3043729 and Berkelbach
https://doi.org/10.1063/1.5032314 to show connections between the
Bethe-Salpeter equation (BSE) formalism combined with the $GW$ approximation
from many-body perturbation theory and coupled-cluster (CC) theory at the
ground- and excited-state levels. In particular, we show how to recast the $GW$
and Bethe-Salpeter equations as non-linear CC-like equations. Similitudes
between BSE@$GW$ and the similarity-transformed equation-of-motion CC method
introduced by Nooijen are also put forward. The present work allows to easily
transfer key developments and general knowledge gathered in CC theory to
many-body perturbation theory. In particular, it may provide a path for the
computation of ground- and excited-state properties (such as nuclear gradients)
within the $GW$ and BSE frameworks.
