Bitcoin is a peer-to-peer payment system proposed by Nakamoto in 2008. The bitcoin backbone protocol has been analyzed 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 must be adopted by all honest miners with high probability. The Prism protocol was recently proposed to dramatically improve the blockchain 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 in synchronous networks without the finite lifespan 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 finite lifespan. The properties take the form of explicit expressions in lieu of order optimal results.