Proof-of-stake (PoS) is a promising approach for designing efficient blockchains, where block proposers are randomly chosen with probability proportional to their stake. A primary concern with PoS systems is the "rich getting richer" phenomenon, whereby wealthier nodes are more likely to get elected, and hence reap the block reward, making them even wealthier. In this paper, we introduce the notion of equitability, which quantifies how much a proposer can amplify her stake compared to her initial investment. Even with everyone following protocol (i.e., honest behavior), we show that existing methods of allocating block rewards lead to poor equitability, as does initializing systems with small stake pools and/or large rewards relative to the stake pool. We identify a geometric reward function, which we prove is maximally equitable over all choices of reward functions under honest behavior and bound the deviation for strategic actions; the proofs involve the study of optimization problems and stochastic dominances of Pólya urn processes, and are of independent mathematical interest. These results allow us to provide a systematic framework to choose the parameters of a practical incentive system for PoS cryptocurrencies.

Bitcoin is the first fully-decentralized permissionless blockchain protocol to achieve a high level of security: the ledger it maintains has guaranteed liveness and consistency properties as long as the adversary has less compute power than the honest nodes. However, its throughput is only 7 transactions per second and the confirmation latency can be up to hours. Prism is a new blockchain protocol which is designed to achieve a natural scaling of Bitcoin's performance while maintaining its full security guarantees. We present an implementation of Prism which achieves a throughput of over 70,000 transactions per second and confirmation latency of tens of seconds on networks of up to 1000 EC2 Virtual Machines.

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Entanglement is a key resource for quantum information processing. A widely used tool for detecting entanglement is entanglement witness, where the measurement of the witness operator is guaranteed to be positive for all separable states and can be negative for certain entangled states. In reality, due to the exponentially increasing the Hilbert-space dimension with respective to the system size, it is very challenging to construct an efficient entanglement witness for general multipartite entangled states. For N -partite Greenberger-Horne-Zeilinger (GHZ)-like states, the most robust witness scheme requires N + 1 local measurement settings and can tolerate up to 1/2 white noise. As a comparison, the most efficient witness for GHZ-like states only needs two local measurement settings and can tolerate up to 1/3 white noise. There is a trade-off between the realization efficiency, the number of measurement settings, and the detection robustness, the maximally tolerable white noise. In this work, we study this trade-off by proposing a family of entanglement witnesses with k (2 ≤ k ≤ N + 1) local measurement settings. Considering symmetric local measurements, we calculate the maximal tolerable noise for any given number of measurement settings. Consequently, we design the optimal witness with a minimal number of settings for any given level of white noise. Our theoretical analysis can be applied to other multipartite entangled states with a strong symmetry. Our witnesses can be easily implemented in experiment and applied in practical multipartite entanglement detection under different noise conditions.

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