But in order to prove that to someone that doesn't have full history, they would have to synchronize it, by downloading all the data and verifying all the trans

But in order to prove that to someone that doesn't have full history, they would have to synchronize it, by downloading all the data and verifying all the transitions. With Merkle tree (identified with its root hash, just as blockchain is identified by hash of its head) it is possible to verify there is a value inside without full synchronization.