The 1/3 bound is for general byzantine behavior. If you're willing to get more specific about the type of behavior you want to tolerate, the bounds are more fle

The 1/3 bound is for general byzantine behavior. If you're willing to get more specific about the type of behavior you want to tolerate, the bounds are more flexible[1] (this link is from the HotStuff people, actually!). For example, a node could tolerate no more than "20% Byzantine faults and 30% alive-but-corrupt faults" In turn, this allows different clients to make different assumptions (one huge advantage PoW has over traditional BFT protocols). Also, w.r.t. linear communication, CBC Casper[2] has the ability to come to consensus on a blockchain with, in the limit (and in the optimal case), a single message per block that is finalized. It uses essentially the same form of pipelining that HotStuff does, but to the extreme (at at the cost of a much longer time for blocks to be finalized). Full disclosure: I work on CBC Casper, so take what I'm saying with a grain of salt. [1] https://dahliamalkhi.wordpress.com/2019/04/24/flexible-byzantine-fault-tolerance/ https://dahliamalkhi.wordpress.com/2019/04/24/flexible-byzan... [2] https://github.com/cbc-casper/cbc-casper-paper/blob/master/cbc-casper-paper-draft.pdf https://github.com/cbc-casper/cbc-casper-paper/blob/master/c...