PoW is sufficient to send payment. But usually if we are sending a payment remotely, we also want a product delivered to us back remotely too right? Typically people use fedex, ups, usps etc. Well what if we could decentralize this shipment? This is what this idea would allow. The blockchain would enforce independent shippers to trustfully deliver the item to you. There are many other possible uses but this is the most apparent to me. For more info read my previous post.
You presented your idea as a different way to do consensus, but it seems like you are really proposing a block chain that can be used to prove that a service was provided.
Hmm maybe it does seem that that I was proposing a consensus method and that is my fault. But the intention is how to incorporate useful work/services into a blockchain to be enforced by the blockchain.
The "second" proof of work is is proving that useful work was completed in addition to proving that consensus work as completed. That is where the "squared" comes in.
So it is kind of a play on words.