Cryptocurrency time series on the Binary Complexity-Entropy Plane: Ranking efficiency from the perspective of complex systems

We report the first application of a tailored Complexity-Entropy Plane designed for binary sequences and structures. We do so by considering the daily up/down price fluctuations of the largest cryptocurrencies in terms of capitalization (stable-coins excluded) that are worth approximately $90\text{\%}$ of the total crypto market capitalization. We focus on the basic elements of price motion, comparing them to the backbone features of a random walk associated with the mathematical underpinnings of the Efficient Market Hypothesis. From the location of each crypto on the Binary Complexity-Entropy Plane (BiCEP), we define an inefficiency score, $I$. The results based on the BiCEP analysis, which we substantiate with statistical testing, indicate that only Shiba Inu (SHIB) is significantly inefficient, whereas the majority of cryptocurrency trading operates in close-to-efficient conditions. Broadly, our $I$-based ranking hints that the design and consensus architecture of a crypto is at least as relevant to efficiency as the features that are usually taken into account in the appraisal of the efficiency of financial instruments, namely canonical fiat money. Lastly, this set of results supports the validity of the binary complexity analysis.