Isn't an Energy-Free-Bitcoin or Cryptocurrency having your lunch and eating it too? If you are going to have a cryptocurrency that consumes little energy (or energy consumption doesn't scale with transactional value), how do you secure the network?
For instance, a cyber coin could publish a hash of their block chain in the New York Times everyday. A sound implementation would secure the chain up to that point.
If you're going to have a trusted authority involved, then having a globally-synchronized blockchain is unnecessary. Users could just check with the trusted authority about whether a transaction is valid (not a double-spend), or even just have the trusted authority maintain balances for them.
The central authority would still choose what ended up in the ledger but they would not be able to rewrite history because of the publicized hashes locking it into place.
I meant to sketch out a centralized way of finalizing a public ledger, so yeah, it isn't decentralized.
I'm massively simplifying the idea, but proof of stake delivers new coins (either at predetermined intervals or all at once during the ICO) out in a pseudo-random way with some weight factors. Its almost like buying lottery tickets. https://en.wikipedia.org/wiki/Proof-of-stake