it is theoretically possible to own money on L2 without owning a UTXO on L1, using rollups. this approach has not been completed yet, but I wouldn't dismiss the possibility. owning a UTXO and going to a custodian is a false choice.
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How does that work?
L2 blocks are compressed into proofs. a sequencer inserts the proofs into L1 and the state of a smart contract on L1 is transitioned. a few rollups have worked out an escape hatch so that users can retrieve funds if the sequencer dies, and the sequencer cannot steal anyone's money. but we still don't have an implementation for a decentralized sequencer. a rollup with an escape hatch is better than a sidechain, but finishing the decentralized sequencer would make it perfect.
rollups are the most popular L2 scaling solution. there is only one out of the top ten rollups that has less money in it than lightning. a few teams are working on implementing rollups on bitcoin. the implementations won't require bitcoin to have a fork.