Any neuroscientists care to weigh in? I'm interested in neuroscience on an amateurish level, and my priors on this are minuscule. I'll also admit to a certain moderately irrational bias towards the quantum consciousness crowd. I have little respect for their objectivity and reason, since they appear, to me, to be pushing scientific ideas almost entirely because they are comforting.
Also, the listed argument doesn't sit right with me. Synaptic connections are impermanent, but there's strong clinical and experimental evidence that memories are explicitly rewritten into new synapses every time they are remembered. This provides an explanation for why they are so fragile and unreliable. I'm not at all sure that the problem they're claiming to solve is actually a problem.
A recent article in PloS Computational Biology suggests that memory is encoded in the microtubules. "Signaling and encoding in MTs and other cytoskeletal structures offer rapid, robust solid-state information processing which may reflect a general code for MT-based memory and information processing within neurons and other eukaryotic cells."
They argue that synaptic connections are transient compared with the lifetime of memories, and therefore memories cannot be stored in them, but in some more persistent structure. The structure they suggest is the phosphorylation state of sites on microtubule lattices within neurons. And that's about as much of the technical detail as I feel able to summarise. It's not all speculation, they report technical work on the structures of these cellular components. Total memory capacity would be somewhere upwards of 10^20 bits (or in more everyday units, 10 million terabytes), depending on the encoding, of which they suggest several schemes.
Journalistic writeup here.
Note that Stuart Hameroff, one of the authors, is known for his proposals for microtubules as the mechanism of consciousness through quantum effects (and with Penrose, quantum gravitational effects). The present paper, however, is solely about memory and does not touch on quantum coherence or consciousness.