Blockchain technology started as the backbone for cryptocurriencies and it has emerged as one of the most interesting technologies of the last decade. It is a new paradigm able to modify the way how industries transact. Today, the industries’ concern is about their ability to handle a high volume of data transactions per second while preserving both decentralization and security. Both decentralization and security are guaranteed by the mathematical strength of cryptographic primitives. There are two main approaches to achieve consensus: the Proof-of-Work based blockchains—PoW—and the Proof-of-Stake—PoS. Both of them come with some pros and drawbacks, but both rely on cryptography. In this survey, we present a review of the main consensus procedures, including the new consensus proposed by Algorand: Pure Proof-of-Stake—Pure PoS. In this article, we provide a framework to compare the performances of PoW, PoS and the Pure PoS, based on throughput and scalability.

For classical matroids, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for q-matroids, the q-analogue of matroids. This is a lot less straightforward than in the classical case, as we will try to convince the reader. With the use of submodular functions and the q-analogue of matroid union we come to a definition of the direct sum of q-matroids. As a motivation for this definition, we show it has some desirable properties.