  To date, no framework combining quantum field theory and general relativity
and hence unifying all four fundamental interactions, exists. Violations of the
Einstein's equivalence principle (EEP), being the foundation of general
relativity, may hold the key to a theory of quantum gravity. The universality
of free fall (UFF), which is one of the three pillars of the EEP, has been
extensively tested with classical bodies. Quantum tests of the UFF, e.g. by
exploiting matter wave interferometry, allow for complementary sets of test
masses, orders of magnitude larger test mass coherence lengths and
investigation of spin-gravity coupling. We review our recent work towards
highly sensitive matter wave tests of the UFF on ground. In this scope, the
first quantum test of the UFF utilizing two different chemical elements, Rb-87
and K-39, yielding an E\"otv\"os ratio $\eta_{\,\text{Rb,K}}=(0.3\pm 5.4)\times
10^{-7}$ has been performed. We assess systematic effects currently limiting
the measurement at a level of parts in $10^8$ and finally present our
strategies to improve the current state-of-the-art with a test comparing the
free fall of rubidium and ytterbium in a very long baseline atom interferometry
setup. Here, a 10 m baseline combined with a precise control of systematic
effects will enable a determination of the E\"otv\"os ratio at a level of parts
in $10^{13}$ and beyond, thus reaching and overcoming the performance limit of
the best classical tests.
