  Resumming quantum fluctuations at the level of the gravitational path
integral is expected to result in non-local effective actions and thus in a
non-trivial momentum dependence of the propagator. Which properties the
(dressed) graviton propagator has to satisfy and whether they can all be met
are key open questions. In this work we present criteria and conditions for the
momentum dependence of a graviton propagator which is consistent with
unitarity, causality, and stability in a non-perturbative setting. To this end,
we revisit several aspects of these conditions, highlighting some caveats and
subtleties that got lost in recent discussions, and spelling out others that to
our best knowledge have not been studied in detail. We discuss the consequences
of these concepts for the properties of the graviton propagator. Finally, we
provide examples of propagators satisfying unitarity and causality, while
avoiding tachyonic and vacuum instabilities, and allowing for an analytic Wick
rotation.
