- Traditional—voting-based consensus
- Lottery-based—Nakamoto and post-Nakamoto consensus
Traditional voting-based consensus has been researched in distributed systems for many decades. Many fundamental results and a lot of ground-breaking work have already been produced in this space. Algorithms like Paxos and PBFT are prime examples of such types of algorithms. Traditional consensus can also be called fault-tolerant distributed consensus. In other words, this is a class of consensus algorithms that existed before Bitcoin and has been part of distributed system research for almost three decades.
Lottery-based or Nakamoto-type consensus was first introduced with Bitcoin. This class can also be simply called blockchain consensus.