  In non-equilibrium Green's function calculations the use of the Generalized
Kadanoff-Baym Ansatz (GKBA) allows for a simple approximate reconstruction of
the two-time Green's function from its time-diagonal value. With this a drastic
reduction of the computational needs is achieved in time-dependent
calculations, making longer time propagation possible and more complex systems
accessible. This paper gives credit to the GKBA that was introduced 25 years
ago. After a detailed derivation of the GKBA, we recall its application to
homogeneous systems and show how to extend it to strongly correlated,
inhomogeneous systems. As a proof of concept, we present results for a
2-electron quantum well, where the correct treatment of the correlated electron
dynamics is crucial for the correct description of the equilibrium and dynamic
properties.
