The "Two-phase Proof of Work" is an evolution of Bitcoin proposed by Emin Gün Sirer to mitigate the risk of very large pools getting too much control on Monero.
A two-phase PoW consists of a block that has two separate cryptopuzzles in it. Under a two-phase PoW:
The double hash of the header (SHA256(SHA256(header))) is smaller than a difficulty parameter X, and The header is signed with the coinbase transaction's private key, and the hash (SHA256(SIG(header, privkey))) of that signature is smaller than a second difficulty parameter Y.
The first phase (i.e. point 1 above) is identical to the existing Bitcoin cryptopuzzle. Our solution retains that mechanism in its entirety. The existing mining rigs are all geared to solve the first puzzle, which they can do very efficiently. X is the value of the current variable known as "difficulty" in the Bitcoin software. So we change absolutely nothing that already exists.
The twist here is that we introduce a second cryptopuzzle. In effect, phase 1 requires our miners to do the work that miners have always been doing. But then, when they think they have a viable solution, we now ask them to sign the block with the private key that controls the payment address, and see if the result, when hashed, is below a second difficulty parameter Y.
This mechanism allows miners to use existing rigs, albeit with a lower difficulty value (X) than the difficulty value currently in effect. This would enable miners to produce a lot more potential solutions; that is, headers that pass phase 1, which we call half-solutions. For each such half-solution, miners use a second device, perhaps a CPU or a specialized card, to perform the second check until a full solution is found.
Is there any technical issue preventing the implementation of a similar mechanism while keeping privacy properties of Monero?