It seems to me that once Casper is fully implemented, the best of both worlds would be for the valuable and scarce BTC token to live on the expedient, efficient Ethereum blockchain. But would this be technically feasible?
Quoted from Vitalik Buterin on Reddit: (https://www.reddit.com/r/ethereum/comments/4d5z1y/is_it_possible_for_bitcoin_to_live_on_the/)
"BTC -> ethereum convertibility can be done fine; the hard part is the ethereum -> BTC conversion. There would need to be some kind of sidechain protocol that can "understand" receipts being created on the ethereum blockchain; this is nontrivial and would have to be done as a softfork and accepted by BTC miners; if you don't want to do this, then the two alternative routes are (1) a federated multisig peg with independent and known parties, and (2) a multisig peg with security deposits on ethereum to incentivize participants to sign transactions correctly. (2) does rely on ETH continuing to have value and not dropping too quickly though, as otherwise the collateral may suddenly be insufficient to provide an adequate incentive not to steal the BTC.
I'm not an expert. But you can implement any currency in Ethereum. So nothing prevents you to do so.
However, everyone (at least every miner) would have to agree at some point and make a migration process. Something which freezes bitcoin network transactions forever, and the only possible transaction is paying TO "someone" who grants the owner of the sender account bitcoins the same amount of "Ether-currency". This kind of change (changing the rules after all) probably needs something like a hard-fork @ bitcoin network too.
And of course, everyone has to trust that "someone" (or at least the mechanism which makes it safe). As he would basically be destroying bitcoins and creating ethereum.
Another interesting discussion would be performance and robustness of both networks (which I do not fully understand).
On chain It's called wrapped BTC if you're just talking a representation of bitcoins value living on the chain. However O.P. is talking about actual bitcoin living on the eth chain, and that's a NO. It has nothing to do with the etherium side. It has to do with the BTC network not being able to see the coins on a non-native network and transaction settlement speed requirements. Think about it. How could bitcoin's native chain prove a btc on the eth chain wasn't double spent at some point before being transferred back to the native chain? You see it as a convenience but in reality it's a potential attack vector. That's why we have oracles and AMMs and apps like change now to streamline conversion processes while still maintaining the integrity of both chains