About 160 years ago, the concept of entropy was introduced in thermodynamics by Rudolf Clausius. Since then, it has been continually extended, interpreted, and applied by researchers in many scientific fields, such as general physics, information theory, chaos theory, data mining, and mathematical linguistics. This paper presents The Entropy Universe, which aims to review the many variants of entropies applied to time-series. The purpose is to answer research questions such as: How did each entropy emerge? What is the mathematical definition of each variant of entropy? How are entropies related to each other? What are the most applied scientific fields for each entropy? We describe in-depth the relationship between the most applied entropies in time-series for different scientific fields, establishing bases for researchers to properly choose the variant of entropy most suitable for their data. The number of citations over the past sixteen years of each paper proposing a new entropy was also accessed. The Shannon/differential, the Tsallis, the sample, the permutation, and the approximate entropies were the most cited ones. Based on the ten research areas with the most significant number of records obtained in the Web of Science and Scopus, the areas in which the entropies are more applied are computer science, physics, mathematics, and engineering. The universe of entropies is growing each day, either due to the introducing new variants either due to novel applications. Knowing each entropy’s strengths and of limitations is essential to ensure the proper improvement of this research field.

Kolmogorov complexity and Shannon entropy are conceptually different measures. However, for any recursive probability distribution, the expected value of Kolmogorov complexity equals its Shannon entropy, up to a constant. We study if a similar relationship holds for R´enyi and Tsallis entropies of order α, showing that it only holds for α = 1. Regarding a time-bounded analogue relationship, we show that, for some distributions we have a similar result. We prove that, for universal time-bounded distribution m^t(x), Tsallis and Rényi entropies converge if and only if α is greater than 1. We also establish the uniform continuity of these entropies.

Quantum Computing (QC) is a fast-growing field that has enhanced the emergence of new programming languages and frameworks. Furthermore, the increased availability of computational resources has also contributed to an influx in the development of quantum programs. Given that classical and QC are significantly different due to the intrinsic nature of quantum programs, several aspects of QC (e.g., performance, bugs) have been investigated, and novel approaches have been proposed. However, from a purely quantum perspective, maintenance, one of the major steps in a software development life-cycle, has not been considered by researchers yet. In this paper, we fill this gap and investigate the prevalence of code smells in quantum programs as an indicator of maintenance issues. We defined eight quantum-specific smells and validated them through a survey with 35 quantum developers. Since no tool specifically aims to detect quantum smells, we developed a tool called QSmell that supports the proposed quantum-specific smells. Finally, we conducted an empirical investigation to analyze the prevalence of quantum-specific smells in 15 open-source quantum programs. Our results showed that 11 programs (73.33%) contain at least one smell and, on average, a program has three smells. Furthermore, the long circuit is the most prevalent smell present in 53.33% of the programs.