One of the aims of this paper is to attract the attention of quantum theorists to certain areas of arithmetic geometryArithmetic geometry whose ideas and concepts and analogues of objects may find applications in quantum physics. Proposing or drawing these interdisciplinary links is novel and may open up new research directions. We discuss several analogies between some developments in arithmetic geometry, once stemming from Grothendieck and now including the IUT theory of Mochizuki, and some aspects of quantum theory including quantum computing. These analogies were spotted recently and it is hoped that related developments may be fruitful. In the appendix we also propose a new abc-ABC question about an asymptotic symmetry of the moduli space of Frey–Hellegouarch elliptic curves over rational numbers. This question goes beyond the standard abc inequalities conjectures/questions. We prove that the positive question to the question and the effective abc inequalities established in [27] using enhanced IUT theory imply the stronger version of the effective \((1+\varepsilon )\) -abc inequality.