Nuclear uncertainties in the production of $p$ nuclei in massive stars have been quantified in a Monte Carlo procedure. Bespoke temperature-dependent uncertainties were assigned to different types of reactions involving nuclei from Fe to Bi. Their simultaneous impact was studied in postprocessing explosive trajectories for three different stellar models. It was found that the grid of mass zones in the model of a 25 $M_odot$ star, which is widely used for investigations of $p$ nucleosynthesis, is too crude to properly resolve the detailed temperature changes required for describing the production of $p$ nuclei. Using models with finer grids for 15 $M_odot$ and 25 $M_odot$ stars with initial solar metallicity, it was found that most of the production uncertainties introduced by nuclear reaction uncertainties are smaller than a factor of two. Since a large number of rates were varied at the same time in the Monte Carlo procedure, possible cancellation effects of several uncertainties could be taken into account. Key rates were identified for each $p$ nucleus, which provide the dominant contribution to the production uncertainty. These key rates were found by examining correlations between rate variations and resulting abundance changes. This method is superior to studying flow patterns, especially when the flows are complex, and to individual, sequential variation of a few rates.
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