This would be a modified version of the
proof of burn system.
Each mining "rig" would generate a stream of hashes, one per second.
Hash(id + timestamp) / burn
The id would be hash(hash of the burning tx + blocks headers for the 100 blocks (including at least 1 POW block) after the coins were burnt)
This means that the hash sequence is fully deterministic (after it has been generated).
Ignoring difficulty changes, each miner would get a set of timestamps where they win a block. Ties could be broken by the lowest hash (but 2 wins in the same second are low probability).
Virtual miners can claim a block by signing the block header, including the timestamp and hash of the previous header.
However, miners would only be allowed to sign a block once for each "win".
The miner's rig would be destroyed if they sign two blocks with the same timestamp. A "proof of cheating" block header could be included into the chain if a rig signs twice for the same second. This sets the POW equivalent for all blocks with the same second to zero and also cancels coinbase/tx fee payout of any blocks that haven't reached coinbase maturity signed by that rig.
If a miner adds a block to a chain and then it is orphaned, then the miner has effectively wasted his low hash value. This creates an incentive to make sure you only sign the chain that will remain as the main chain.
You need to make sure that the total (real/non-virtual) POW for system is always much higher than the total value of all the coins. The value can be estimated based on POW difficulty = (total coins) * (hashes per block) / (block reward). The virtual mining rig target would be set to keep one block per 10 mins, but the POW target could be set to keep sufficient total POW value. However, using difficulty as an input into calculating difficulty could cause positive feedback.
One issue, in general, is that while the coin's value stays constant, then there is a threshold difficulty. If the difficulty is to low, you get lots of hashing (since the coin has the best ROI). OTOH, if it is to high, then you get zero.
Bitcoin doesn't have the issue, since it is the main market for hashing (if its difficulty increases, then demand drops, and that has an effect on market price for hashing). Small alt-coins are closer to pure-competition for hashing, so they can cross a sudden threshold and go from lots of hashing to almost none.
