In recent years, the field of quantum technologies has experienced significant growth. During this expansion, quantum simulators have emerged as one of the most experimentally advanced platforms. These devices can simulate highly complex quantum systems. Moreover, they provide an area of research in which Europe currently holds a competitive advantage. There is a growing consensus among physicists that quantum simulators have already achieved a so-called quantum advantage. However, these results have received considerably less attention and scrutiny than similar claims made for digital quantum computers. TouQan aims to bridge the gap in our understanding of potential quantum advantages in quantum simulators. We do this by engaging in a "cat-and-mouse" investigation, in which each of the objectives aims to narrow down the scope of the others. Firstly, we will advance our knowledge of the computational power of simulators from a mathematically rigorous perspective and will find ways of characterizing it, including the impact of hardware noise. Secondly, we will examine under which conditions classical computers can effectively simulate quantum simulators to narrow down when they do not offer significant advantages. By doing so, a clearer picture of these devices' computational power will emerge, attracting a wider community of researchers to quantum simulators and strengthening Europe's position in the sector. The interdisciplinary nature of this project will require the use and development of advanced tools at the intersection of computer science, physics, and mathematical physics. Our consortium comprises five ambitious early-career researchers with excellent track records based in various European countries. This collaborative effort will foster increased cooperation among European nations on a timely topic, ultimately bolstering the development of quantum simulators and enhancing our understanding of their computational power and, in particular, of how they offer the widely-sought quantum advantage.