Where is the lightning history persisted? In relation to this question:

Alice &amp; Bob where they transfer 1 BTC each other.


Both Alice &amp; Bob transfer 1 BTC into a shared multisig address. 
A ‘balance sheet’ is created where 1 BTC is paid by Alice -> Bob and 1 BTC is paid by Bob -> Alice. 
In the above transaction, Alice signs the transaction and gives to Bob and vice-versa.  
Either Alice or Bob can sign their counterparts payout transaction and broadcast it to the blockchain. 
Alice will update the ‘Balance sheet’ as 0.5 BTC to Alice and 1.5 to Bob and send the payout transaction to Bob to sign.
To ensure neither Alice  or Bob revert to the old transaction, each will create an ‘anti-cheat’ transaction.  
If Alice and Bob wants to close the channel, they can consensually close the channel and broadcast the transaction to the blockchain.


Where is this balance sheet persisted? What happens if BOTH Alice AND Bob lose this information?
We like to say that Bitcoin is decentralized, and it is, but all the data can be viewed as having a centralized "source of truth": the blockchain.  All transactions that have ever occurred are a part of this source of truth.  The management and upkeep of this source of truth, however, is decentralized.  Nobody is in charge of it, yet everybody agrees on a single set of data.

The lightning network, however, is even more decentralized.  It doesn't just manage and update itself without central authority, but there is no agreed upon "truth" of the network.  The activity that occurs between two participants in a channel is only the business of the participants, and nobody else ever needs to see it.  Therefore it is not shared, and unless the participants want to, it is not stored.  Once a channel closes, the participants have completely settled all transactions between each other with a single transaction to the main Bitcoin blockchain.  They may very well persist the activity data for their own purposes, but it is in no way required for them to continue to participate.