Extreme-ultraviolet observations of warm coronal loops suggest that they are bundles of unresolved strands that are heated impulsively to high temperatures by nanoflares. The plasma would then have the observed properties (e.g., excess density compared with static equilibrium) when it cools into the 1-2MK range. If this interpretation is correct, then very hot emission should be present outside of proper flares. It is predicted to be very faint, however. A critical element for proving or refuting this hypothesis is the existence of hot, yet faint plasmas which should be at amounts predicted by impulsive heating models. We report on the first comprehensive spectroscopic study of hot plasmas in active regions (ARs). Data from the Extreme-ultraviolet Imaging Spectrometer on Hinode were used to construct emission measure (EM) distributions in quiescent ARs in the 1-5 MK temperature range. The distributions are flat or slowly increasing up to approximately 3 MK and then fall off rapidly at higher temperatures. We show that AR models based on impulsive heating can reproduce the observed EM distributions relatively well. Our results provide strong new evidence that coronal heating is impulsive in nature.