Proof-of-Stake blockchains based on a longest-chain consensus protocol are an attractive energy-friendly alternative to the Proof-of-Work paradigm. However, formal barriers to getting the incentives right were recently discovered, driven by the desire to use the blockchain itself as a source of pseudorandomness . We consider instead a longest-chain Proof-of-Stake protocol with perfect, trusted, external randomness (e.g. a randomness beacon). We produce two main results. First, we show that a strategic miner can strictly outperform an honest miner with just 32.8% of the total stake. Note that a miner of this size cannot outperform an honest miner in the Proof-of-Work model . This establishes that even with access to a perfect randomness beacon, incentives in Proof-of-Work and Proof-of-Stake longest-chain protocols are fundamentally different. Second, we prove that a strategic miner cannot outperform an honest miner with 30.8% of the total stake. This means that, while not quite as secure as the Proof-of-Work regime, desirable incentive properties ofProof-of-Work longest-chain protocols can be approximately recovered via Proof-of-Stake with a perfect randomness beacon. The space of possible strategies in a Proof-of-Stake mining game is significantly richer than in a Proofof- Work game. Our main technical contribution is a characterization of potentially optimal strategies for a strategic miner, and in particular a proof that the corresponding infinite-state MDP admits an optimal strategy that is positive recurrent.