There are some interesting papers working on BFT problems for proof-of-stake blockchains now. I’ll mostly refrain from linking those, but instead highlight some building blocks that themselves do not require stake, slashing, chains, etc.
There is a remarkably simple not-actually-a-blockchain protocol from UCL folk that ensures nodes broadcasts the same messages to all other nodes, but does use slashing somewhere: https://arxiv.org/pdf/1809.01620.pdf It's similar to the slashing free scheme https://dspace.mit.edu/bitstream/handle/1721.1/107927/MIT-CSAIL-TR-2017-004.pdf but with many senders. Also, there is an older paper also by Silvio Micali that provides a more general agreement scheme: https://people.csail.mit.edu/silvio/Selected%20Scientific%20Papers/Distributed%20Computation/BYZANTYNE%20AGREEMENT%20MADE%20TRIVIAL.pdf Both of these last two are components of Algorand, btw. There are also interesting things being built with verifiable secret sharing (VSS) and publicly verifiable secret sharing (PVSS). In brief the difference goes, there is extra protocol complexity VSS to manage some accusation phase since the shares’ correctness can only be verified by the recipients, but VSS agrees on scalars so the result can be used to produce many signatures or whatever faster, while PVSS agrees on curve points, like maybe a BLS signature, which each require more of the PVSS protocol. There were bugs in previous VSS protocols, but they appear to all be fixed here: https://link.springer.com/content/pdf/10.1007%2F3-540-48910-X_21.pdf Best, Jeff
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