Now thinking about it, you can probably fully sync and validate the time chain without actually seeing any of the full transactions.

If you have a prover that converts transactions into zk proofs, revealing only unspent outputs, tx hashes (for block hash / pow verification).

You can sync to current utxo set without seeing any of the full txs and their scripts and without trusting anyone, you just follow longest pow proofchain.

I don't think it's very useful (harder to verify zk proofs than the timechain itself). It does not add privacy because the prover sees all the txs. But it's mind-blowing that it can probably be done.