Blockchains (e.g. Bitcoin, Algorand, Byzcoin, Hyperledger, RedBelly etc) became a game changer in the distributed storage area due to their ability to mimic the functioning of a classical traditional ledger such as transparency and falsification-proof of documentation in an untrusted environment where the computation is distributed, the set of participants to the system are not known and it varies during the execution. However, the massive integration of distributed ledgers in industrial applications strongly depends on the formal guaranties of the quality of services offered by these applications, especially in terms of consistency. Our work continues the line of recent distributed computing community effort dedicated to the theoretical aspects of blockchains. This paper is the first to specify the distributed shared ledgers as a composition of \emph{abstract data types} all together with an hierarchy of \emph{consistency criteria} that formally characterizes the histories admissible for distributed programs that use them. Our work extends the consistency criteria theory with new consistency definitions that capture the eventual convergence process in blockchain systems. Furthermore, we map representative existing blockchains from both academia and industry in our framework. Finally, we identify the necessary communication conditions in order to implement the new defined consistency criteria.