The security of most existing cryptocurrencies is based on a concept called Proof-of-Work, in which users must solve a computationally hard cryptopuzzle to authorize transactions ("one unit of computation, one vote"). This leads to enormous expenditure on hardware and electricity in order to collect the rewards associated with transaction authorization. Proof-of-Stake is an alternative concept that instead selects users to authorize transactions proportional to their wealth ("one coin, one vote"). Some aspects of the two paradigms are the same. For instance, obtaining voting power in Proof-of-Stake has a monetary cost just as in Proof-of-Work: a coin cannot be freely duplicated any more easily than a unit of computation. However some aspects are fundamentally different. In particular, exactly because Proof-of-Stake is wasteless, there is no inherent resource cost to deviating (commonly referred to as the "Nothing-at-Stake" problem). In contrast to prior work, we focus on incentive-driven deviations (any participant will deviate if doing so yields higher revenue) instead of adversarial corruption (an adversary may take over a significant fraction of the network, but the remaining players follow the protocol). The main results of this paper are several formal barriers to designing incentive-compatible proof-of-stake cryptocurrencies (that don't apply to proof-of-work).