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Quantum Processes in strong classical potentials are ubiquitous problems spanning multiple areas of physics from QED over QCD to gravity. Examples include Hawking radiation, cosmological particle production, thermalization of quark-gluon plasmas, the proton spin crisis and vacuum polarization and pair production.In QED, the long-standing challenge to describing charged particle dynamics in strong classical electromagnetic fields is how to incorporate classical radiation, classical radiation reaction and quantized photon emission into a consistent unified framework. The current, semi-classical methods to describe the dynamics of quantum particles in strong classical fields also provide the theoretical framework for fundamental questions in gravity and QCD. However, as we show, these methods break down for highly relativistic particles propagating in strong fields. They must therefore be improved and adapted.As controlled experiments involving black holes or controlling and resolving in detail a hadron-hadron collision pose a certain degree of technical challenge, quantum electrodynamics poses the best chance to develop a full Strong-Field Quantum Field Theory. Utilizing the newest generation of ultrahigh intensity lasers in combination with accelerators we now have the technology to enable a dedicated experimental effort, offering the best controllable context to establish a robust, experimentally validated foundation for the fundamental theory of quantum effects in strong classical potentials.Once such a full Strong-Field Quantum Field Theory is tested and validated for QED, the methods can be transferred to other areas such as QCD. What is more, with the control of ultrahigh intensity laser-particle interactions it will potentially become possible to directly test aspects of other areas of physics in new regimes, such as the gravity-inertia equivalency or parton distribution functions in e-/A collisions. Even BSM physics like the search for dark photons could become possible if high enough particle energy and particle flux can be achieved by laser-driven accelerators.How could such a project be realized within the context of new facilities like ELI-NP and other upcoming or proposed facilities and why now?Ultrahigh intensity lasers and there interactions with particles and matter are at the forefront of research and development and are poised to open up new regimes of physics at the intensity frontier either directly or by enabling advances in particle acceleration. The recent Nobel Prize in physics in 2018 for D. Strickland and G. Mourou speak to their potential. Using examples of the first ultrahigh intensity experiments above I>1022 W/cm2, at the Texas Petawatt Laser and the 4PW laser at the Center for Relativistic Laser Science I will outline lessons learned and requirements for a systematic push towards Quantum Field Physics with Ultrahigh Intensity lasers.