Future planned space telescopes, such as the IR/O/UV Large Telescope, recommended by Astro2020 will be used to directly image exo-Earths. They will employ high-order wavefront sensing and control (HOWFSC) to correct static and slow wavefront errors in the image plane to achieve contrasts better than 10⁹. Our work evaluates the computational requirements for HOWFSC algorithms and compares these to the capabilities of processors that are expected to be available during mission development. We find that HOWFSC creates unprecedented requirements for space-based computational power, such as the ∼10¹³ floating-point operations necessary to generate the dark hole, based on the Large UV/Optical/IR (LUVOIR) study. In our worst-case estimates, maintaining an LUVOIR-size dark hole at 10¹⁰ contrast might require up to several orders of magnitude more computational throughput than available on the most advanced radiation-hardened processor.