In the context of the Drivechain proposal, for UTXOs in miner-controlled escrows to be spent, they have to be voted on by each block over a large number of blocks, right?

Now, how are these votes made and where are they computed? I think the votes are somehow written in the block coinbase, but how exactly? It's not like there's much space there. And then each full node has to keep a table of what transaction has how many votes and verify against that whenever a transaction is included in the block?

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The votes are published on the block coinbase. This can be done in multiple ways, some more efficient than others. The BIP-300 specification proposes a way that is somewhat complex but tries to be efficient.

The actual current count at any point -- i.e. the sum of all upvotes and downvotes -- is computed from these coinbase votes and kept by all nodes as a part of the chainstate. Again, the actual physical place of where these things are kept is an implementation detail (they could be either kept in a separate table or somewhere associated with the UTXO that is holding the Drivechain funds).

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