There's also proof-of-burn in the context of bidding for the right of block validation, though that's far less developed than proof-of-share.
If you drop the distributed requirement you can run an equivalent system on on a pentium 4 in someone's basement, without wasting GWs of power and spewing carbon dioxide into the atmosphere.
Huh? Exponentially? The difficulty level is rising exponentially, but that's owing to continued investment in specialized hashing hardware. I certainly don't think that the actual amount of electricity being put into hashing is still growing exponentially. It would only grow exponentially if the price of Bitcoin continued to appreciate exponentially, and that hasn't been the case in years.
Gasoline at least gets you from A to B. Burning cycles is much more wasteful. Bitcoin's proof of work/waste is really something like 'I can prove that it is statistically likely that I wasted a shit-ton of power heating this data centre, doing SHA hashes, trying to find one that has a high number of leading zeros.' That's pretty wasteful IMO.
One of the obvious things about Merkle ledgers is that everyone can have their own. You could have one per tradeable instrument on an exchange, for example. Anyone can start an altcoin; why not the BoE?
The proof-of-waste system allows the system to be fully distributed by continuously burning electricity such as to push up the minimum cost of overwriting some other node's choice of doublespend (note: NOT ability to forge transaction!) to very high levels.
If you discard the "distributed" requirement, you can save a lot of electricity by having a single or multiple CA-like series of authorized stamp-issuers. It then looks a lot more like Chaum's digital cash.
If you discard "pseudonymous" as well, noting that bitcoin pseudonymity is both fragile and directly contrary to KYC/AML, you can make further optimisations.
signature verification, P2P, cryptographic hashing, public-key cryptography
Old, but not yet widely deployed outside of the browser-CA system. Getting banks to use proper digital signature technology would be a major upgrade.
(Also, remember in the UK we have both EMV and "faster payments", so the US problems of stolen magswipes and waiting for cheques are not so prevalent.)