  We present in detail the scientific objectives in fundamental physics of the
Space-Time Explorer and QUantum Equivalence Space Test (STE-QUEST) space
mission. STE-QUEST was pre-selected by the European Space Agency together with
four other missions for the cosmic vision M3 launch opportunity planned around
2024. It carries out tests of different aspects of the Einstein Equivalence
Principle using atomic clocks, matter wave interferometry and long distance
time/frequency links, providing fascinating science at the interface between
quantum mechanics and gravitation that cannot be achieved, at that level of
precision, in ground experiments. We especially emphasize the specific strong
interest of performing equivalence principle tests in the quantum regime, i.e.
using quantum atomic wave interferometry. Although STE-QUEST was finally not
selected in early 2014 because of budgetary and technological reasons, its
science case was very highly rated. Our aim is to expose that science to a
large audience in order to allow future projects and proposals to take
advantage of the STE-QUEST experience.
