I bump this olde article because I found it only today and I consider it far better written then a recent article about the same topic on Coindesk. Good work!

I was curious if Proof of Activity was ever implemented, it looks interesting. The other algorithms now have all been used for altcoins:

- Proof of Capacity: Burst
- Proof of Burn: Slimcoin and the defunct TGCoin (Third Generation Coin)
- Hybrid Proof of Work/Stake: Peercoin and a lot of clones
- Byzantine Generals: Bitshares, Lisk and other "DPOS" coins; Ethereum's planned "Casper" is based on a similar idea but uses an algorithm where the participants are not limited to a closed group like in DPOS.

There are two algorithms missing:

- Transactions as Proof of Stake (TAPOS): was never fully implemented (afaik) but parts of the concept are used in Bitshares and Nxt.
- Proof of Importance: Proof of Stake weighted by a transaction graph (NEM)