The balance-blockchain (=ledger-blockchain ?) would work the same as the transaction-blockchain.
The concept of a blockchain can be applied to anything.
This includes all the blockchain-mechanisms for rejecting and accepting, etc.
So why should we trust a balance-blockchain? Should we also have a balancechain for that, and then another for that?
The real question is why not ?
You can change the balance, but it will not be accepted by other clients, because your balance would be a mismatch.
The mismatch can happen in two ways:
1. The traditional way by: length wins, which is what bitcoin does.
2. The voting way: confirmations.
An attacker could create a fake balance chain, which could be longer, and broadcast it to you.
However this should not happen because of majority of processing power, so majority has ultimately longer balance chain.
Unless you are disconnected from the real network and you connect to a fake network, and you cannot connect to real network, same problem as with transaction block chain.
Other attacks could exist too.
The hashes in the balance chain also protect against modifications, make a single modification and you will have to find a hash collision, or re-do an entire fake balance chain which becomes computationally expensive. A balance chain might not even be necessary. Only one balance sheet needs to be stored the previous one. However balance sheets could also function as checkpoints, but that's probably not necessary.
My suggestion was also to make the difficulty 1000x greater than the blockchain, if the blockchain is allowed to be 1000 blocks ahead, if this leads to 1000x hash search times remains to be seen, maybe it will become too difficult, examination of this idea will have to be done.
So the concept of a balance-chain could be dismissed, instead a single balance sheet is stored by all clients.
It could be considered a "sliding balance sheet". It slides along the transactions over time.
However if the concept of a balance chain is dismissed, then the race can no longer happen, nobody wins.
This becomes a problem, there is no winning selection criterium anymore, except perhaps the voting situation. Though that's a bit shakey.
Therefore the idea could be simply:
Store the hashes of the balance chain only.
The balance sheets themselfes are thrown away, except the last one.
At least this makes it possible for people who have all transactions and know exactly when to calculate a balance and a balance hash to check the chain for consistency.
This also preserves the proof-of-work idea.
An attacker would have to re-do those balance chain hashes, computationally expensive.
It's even very interesting, since an attacker might not have the necessary data to make the balance chain.
Therefore attacker is restrained to new data only, less attack surface for him. This would be a lazy attacker, however it's good to assume default intense attackers which have all data.
By storing balance chain hashes only, the old transactions can be thrown away, the old balance sheets can be thrown away, what remains are the balance chain hashes, about 256 bits per hash and a recent balance sheet.