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Advanced AO systems will likely utilise Pyramid wave-front sensors (PWFS)over the traditional Shack-Hartmann sensor in the quest for increasedsensitivity, peak performance and ultimate contrast. Here, we wish to bringknowledge and quantify the PWFS theoretical limits as a means to highlight itsproperties and use cases.  We explore forward models for the PWFS in the spatial-frequency domain forthey prove quite useful since a) they emanate directly from physical-optics(Fourier) diffraction theory; b) provide a straightforward path to meaningfulerror breakdowns, c) allow for reconstruction algorithms with $O (n\,log(n))$complexity for large-scale systems and d) tie in seamlessly with decoupled(distributed) optimal predictive dynamic control for performance and contrastoptimisation. All these aspects are dealt with here. We focus on recentanalytical PWFS developments and demonstrate the performance using bothanalytic and end-to-end simulations.  We anchor our estimates with observed on-sky contrast on existing systems andthen show very good agreement between analytical and Monte-Carlo estimates forthe PWFS. For a potential upgrade of existing high-contrast imagers on10\,m-class telescopes with visible or near-infrared PWFS, we show under medianconditions at Paranal a contrast improvement (limited by chromatic andscintillation effects) of 2x-5x by replacing the wave-front sensor alone atlarge separations close to the AO control radius where aliasing dominates, andfactors in excess of 10x by coupling distributed control with the PWFS overmost of the AO control region, from small separations starting with the InnerWorking Angle of typically 1-2 $\lambda/D$ to the AO correction edge (here 20$\lambda/D$).

Performance limits of adaptive-optics/high-contrast imagers with pyramid wave-front sensors