Suppose I do a transaction with an UXTO created years ago. Would that take longer to confirm because miners would have to look further back in the transaction chain to verify it?
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No.
The miners use Bitcoin Core or similar full node software to determine which transactions are valid and suitable for inclusion in a block. Bitcoin Core maintains a database of all unspent UXTOs. This database is constructed when the node first downloads the blockchain and is updated as new blocks and transactions arrive. It's random access and any transaction, regardless of age, can be found by its ID in a very short time.
Nate Eldredge
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How many unspent UXTOs are there currently? – Geremia May 13 '17 at 03:11
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1@Geremia: See https://bitcoin.stackexchange.com/questions/35980/how-big-is-the-utxo-database. I don't have a full node running to get a current figure. – Nate Eldredge May 13 '17 at 04:09