  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 people's 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.
