Recent advances in the blockchain research have been made in two important directions. One is refined resilience analysis utilizing game theory to study the consequences of selfish behaviors of users (miners), and the other is the extension from a linear (chain) structure to a non-linear (graphical) structure for performance improvements, such as IOTA and Graphcoin. The first question that comes to peoples minds is what improvements that a blockchain system would see by leveraging these new advances. In this paper, we consider three major metrics for a blockchain system: full verification, scalability, and finality-duration. We { establish a formal framework and} prove that no blockchain system can achieve full verification, high scalability, and low finality-duration simultaneously. We observe that classical blockchain systems like Bitcoin achieves full verification and low finality-duration, Harmony and Ethereum 2.0 achieve low finality-duration and high scalability. As a complementary, we design a non-linear blockchain system that achieves full verification and scalability. We also establish, for the first time, the trade-off between scalability and finality-duration.
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