Proof of Work solves that by preventing Sybil attacks - it's costly to mine (which is equivalent to signing off on, because it wouldn't make sense to sign an invalid block, you would get rewarded). It's the only signature scheme known to be dynamic and allow for an arbitrary participant size. The point is that you can trust the chain with the most proof of work. As long as you maintain a diverse set of peers and validate blocks, you can know the state of the network.