The high-level idea of ZeroSync is to make it trivial to verify the chain state. Even on phones. However, that proof does not contain the UTXO set, but only a commitment to the UTXO set. The most simple UTXO set commitment would be Utreexo. A more sophisticated approach is to use something like a Merkle-Patricia Tree, which allows you to query all UTXOs of a particular address. This way light clients can sync by downloading less than a megabyte of data. Of course, that requires some kind of bridge nodes, which serve these inclusion proofs to the light clients. Moreover, in this model, clients sacrifice their privacy.

Still, that's the baseline idea and from there we can improve the properties with different tradeoffs. E.g. downloading more data than what you really need can give you much better privacy