If you have a trusted CH(n), you can use it to query prior state transitions in blocks n - 1 and lower from an untrusted full node. The full node will be unable to lie to you, short of forcing a cryptographic hash collision.
SNV is used to prove that specific state transitions occurred in the past. For example, if I want to convince you that I sent a
NAME_UPDATE transaction at block 400000, and you have the consensus hash of block 460000, you can use SNV to verify my claim.
We're still working on updating SNV to prove name ownership. If Alice wanted to prove to Bob that she transferred her name to Charlie at block 430000, Bob can use CH(460000) to verify that the transfer happened. However, he cannot yet prove that it was the last name operation (i.e. he can't prove that Charlie did not transfer the name again). Bob would also need a trusted UTXO source to prove that Charlie did not issue any subsequent transaction that consumed the "name ownership" UTXO embedded within the
We're still working on an efficient way to rely only on the consensus hash to prove who owns which name at each block, so users won't need to build a full UTXO index to use SNV.