  We propose a scheme that generalizes the loss scaling properties of
twin-field or phase-matching quantum key distribution (QKD) related to a
channel of transmission $\eta_{total}$ from $\sqrt{\eta_{total}}$ to
$\sqrt[2n]{\eta_{total}}$ by employing n-1 memory stations with spin qubits and
n beam-splitter stations including optical detectors. Our scheme's resource
states are similar to the coherent-state-based light-matter entangled states of
a previous hybrid quantum repeater, but unlike the latter our scheme avoids the
necessity of employing 2n-1 memory stations and writing the transmitted optical
states into the matter memory qubits. The full scaling advantage of this
memory-assisted phase-matching QKD (MA-PM QKD) is obtainable with threshold
detectors in a scenario with only channel loss. We mainly focus on the
obtainable secret-key rates per channel use for up to n=4 including memory
dephasing and for n=2 (i.e., $\sqrt[4]{\eta_{total}}$-MA-PM QKD assisted by a
single memory station) for error models including dark counts, memory dephasing
and depolarization, and phase mismatch. By combining the twin-field concept of
interfering phase-sensitive optical states with that of storing quantum states
up to a cutoff memory time, distances well beyond 700 km with rates well above
$\eta_{total}$ can be reached for realistic, high-quality quantum memories (up
to 1s coherence time) and modest detector efficiencies. Similarly, the standard
single-node quantum repeater, scaling as $\sqrt{\eta_{total}}$, can be beaten
when approaching perfect detectors and exceeding spin coherence times of 5s;
beating ideal twin-field QKD requires 1s. As for further experimental
simplifications, our treatment includes the notion of weak nonlinearities for
the light-matter states, a discussion on the possibility of using homodyne
detectors, and a comparison between sequential and parallel entanglement
distributions.
