If you actually discuss with people working on quantum memories, you will notice (at least I did) that they share a vague definition : "a quantum memory is something which stores a quantum state" better than a classical memory could do. Beyond that, they have vastly different ideas on
- how to implement a quantum memory (single qubits, collective degrees of freedom, array of qubits impelmenting a topological error correction code ...)
- what to do with a quantum memory (RAM for a quantum computer, store states to reconstruct them later, store states to measure them later )
- how to evaluate the quality of a quantum memory (fidelity, quantum capacity, cheating probability in a noisy-storage model based quantum cryptography protocol...)
Note that the same kind of differences also apply on classical memories, between a sheet of paper, a magnetic tape, an ECC RAM or a group of neurons in my brain.
I'm convinced however that it is possible to give a generic definition of a quantum memory. In a paper (shameless plug) on a specivic kind of continuous variable quantum memory, I wrote
A quantum memory, by definition, stores informations about a quantum state for a given time interval, and it should do it better than any classical memory (i.e. classical-states based memory). Since an a priori known quantum state has a complete classical description (its density matrix), it can be reconstructed with an arbitrarily high fidelity by a setup only storing this description in a classical memory.
More specifically, following the noisy storage model literature, a quantum memory can be defined by a quantum channel, which itself can be described by a time-dependent completely positive (CP) map $\mathcal T_t$. If the quantum memory has a classical-output (e.g. if it is used for a delayed measurement), it can be modelled by a CP map followed by a measurement. It gives us a straightforward criterion to distinguish a classical memory from a quantum memory, since a classical memory supplemented by measurement and preparation can only implement entanglement breaking channel.
If the memory output is quantum, it can be said to be quantum memory iff $\mathcal T_t$ is not entanglement breaking. If the output is classical, one has to show that the outputs cannot be obtained by direct measurements of the input state.
The question whether the memory uses a (cloud of) atom(s) to store a photonic qubit or topologic error correcting codes to store the state of the nuclear spin of 5 NV-centres is irrelevant for the definition. In the same way that the RAM of my computer differs vastly from a poetry book. Both are classical memories.
Then, by definition, the classical capacity cannot be a figure of merit to characterize a quantum memory. But many figures of merits are possible, depending on the application, as with quantum channels. The quantum capacity seems a natural figure of merit, but a memory storing bound entangled state would be excluded by this figure of merit.