  Bitcoin is a peer-to-peer payment system proposed by Nakamoto in 2008.
Properties of the bitcoin backbone protocol have been investigated in some
depth: the blockchain growth property quantifies the number of blocks added to
the blockchain during any time intervals; the blockchain quality property
ensures the honest miners always contribute at least a certain fraction of the
blockchain; the common prefix property ensures if a block is deep enough, it
will eventually be adopted by all honest miners with high probability.
Following the spirit of decoupling various functionalities of the blockchain,
the Prism protocol is proposed to dramatically improve the throughput while
maintaining the same level of security. Prior analyses of the bitcoin and Prism
backbone protocols assume the lifespan of blockchain is finite. This paper
presents a streamlined and strengthened analysis without the finite horizon
assumption. Specifically, the results include a blockchain growth property, a
blockchain quality property, and a common prefix property of the bitcoin
backbone protocol, as well as the liveness and persistence of the Prism
backbone protocol regardless of whether the blockchains have a infinite
lifespan. We also express the properties of bitcoin and Prism backbone
protocols in explicit expressions rather than order optimal results, which lead
to tighter bounds and practical references for public transaction ledger
protocol design.
