  Quantum memories are enabling devices for extending the reach of quantum key
distribution (QKD) systems. The required specifications for memories are,
however, often considered too demanding for available technologies. One can
change this mindset by introducing memory-assisted
measurement-device-independent QKD (MDI-QKD), which imposes less stringent
conditions on the memory modules. It has been shown that, in the case of {\em
fast} single-qubit memories, we can reach rates and distances not attainable by
single no-memory QKD links. Single-qubit memories, such as single atoms or
ions, have, currently, too slow of an access time to offer an advantage in
practice. Here, we relax that assumption, and consider ensemble-based memories,
which satisfy the main two requirements of having short access times and large
storage-bandwidth products. Our results, however, suggest that the
multiple-excitation effects in such memories can be so detrimental that they
may wash away the scaling improvement offered by memory-equipped systems. We
then propose an alternative setup that can in principle remedy the above
problem. As a prelude to our main problem, we also obtain secret key generation
rates for MDI-QKD systems that rely on imperfect single-photon sources with
nonzero probabilities of emitting two photons.
