A major goal of modern physics is to understand and test the regime where quantum mechanics and general relativity both play a role. However, its new effects are thought to be relevant only at high energies or in strong gravitational fields,  beyond the reach of present-day experiments. Here I discuss a novel approach to this challenge, focused on low-energy but composite quantum systems. The key insights is that quantum coherence of composite systems can be measurably affected by time dilation even at low-energies and in weak gravitational fields. I will explain the resulting new effects and how they can be tested. I will particularly focus on new insights about the formulation and validity of the Einstein Equivalence Principle (EEP) from this approach, and sketch some ideas for tests of genuine quantum aspects of the EEP.