can I use elements of the blockchain to protect the integrity of a leaderboard without central servers?
In general, as mentioned in the comments. Your problem in these games resemble a Byzantine Generals Problem where you may experience non-compliant/malicious actors in the network. To achieve consensus in this context, there are multiple avenues.

Firstly, there's Bitcoin's approach. Bitcoin enables every full node to perform independent verification of the Blockchain. The Blockchain is simply a store of a series of past events leading to the current state. Where "past events" (i.e. transactions) are stored in Blocks and the "chain" refers to the hard requirement that each subsequent block is required to be dependent on the previous block based on a set of requirements. Access to the write permissions to the Blockchain is governed by means of Proof-of-Work, where you must burn CPU cycles to create a new block which implicitly grants you a write permission to append to the Blockchain. A key difference here is that the Proof-of-Work need not be done in a collaborative manner as it is a Poisson Process, meaning my chances are just as good as anyone else's, assuming that everyone has the same hash power. In other election algorithms, you may have to negotiate collaboratively and it is not necessarily fully byzantine-failure tolerant. 

So how can some elements of this be applied in your gaming context is left for you to decide. But in principle, yes, elements of Bitcoins are applicable in the context you've described, but again I'd reiterate, it is somewhat "overkill". Maybe in the context of the game, you'd set the leaderboard to have some condition that requires some proof of some action that proves that the player has actually done something to mutate the leaderboard? Otherwise, it'd probably be simpler to do it in Lockstep as in RTS games.