The thing I never understood with private blockchain tech is that the "traditional" blockchain (i.e. Bitcoin) relies on proof of work, and the only way this is viable is to have tons of resources working on these proofs so that you don't get a 51% attack (there have even been a bunch of articles about how smaller coins actually are very susceptible to a 51% attack by a decently funded attacker).
For a private blockchain, though, it never made sense to me as to who would serve the role of the miners with sufficient incentive to prevent a nefarious attacker. If on the other hand you are in a system where the participants agree as to how they will trust each other, well then you'd be back to a situation where the byzantine model isn't really necessary and you can just go back to a cryptographically signed ledger a la the Quantum Ledger DB that AWS just announced.
Would really appreciate someone explaining this one to me!