The interplay of correlations and quantum fluctuations in condensed matter can give rise to topological phases with unexpected and exciting properties. While originally proposed for fractional quantum Hall states, recently new opportunities arose for realizing and controlling topological order: Quantum computers have demonstrated the fascinating fractionalized statistics of topological excitations and moiré semiconductors have appeared as promising candidates for realizing topological order. However, it remains an important open challenge to understand the dynamical response of such entangled matter, both on the fundamental level as well as for providing key experimental signatures that characterize these phases. The central focus of the project DynaQuant is to develop new concepts and new theoretical methods to study the dynamical response of topological quantum states. The project has three principal objectives each of which would represent a major contribution to the field: (O1) To introduce new dynamical probes tailored toward emerging experimental platforms that enable the detection of unique signatures of equilibrium phases with topological order. (O2) To demonstrate the response of pristine nonequilibrium phases with Floquet topological order that do not possess analogues in thermal equilibrium. (O3) To develop novel tensor network approaches for fracton topological order and investigate the collective dynamics of their excitations. To successfully meet our ambitious objectives, my team and I will develop complementary analytical and numerical approaches. This allows us to understand fundamental dynamical properties of entangled quantum matter and to guide future experiments. Due to the international effort in developing experimental platforms for realizing topological order, it is now the right time to foster a deep understanding of their dynamical response, which is the central goal of the project DynaQuant.