  The $s$-process in massive stars produces the weak component of the
$s$-process (nuclei up to $A \sim 90$), in amounts that match solar abundances.
For heavier isotopes, such as barium, production through neutron capture is
significantly enhanced in very metal-poor stars with fast rotation. However,
detailed theoretical predictions for the resulting final $s$-process abundances
have important uncertainties caused both by the underlying uncertainties in the
nuclear physics (principally neutron capture reaction and $\beta$-decay rates)
as well as by the stellar evolution modeling. In this work, we investigated the
impact of nuclear-physics uncertainties relevant to the $s$-process in massive
stars. Using a Monte-Carlo based approach, we performed extensive nuclear
reaction network calculations that include newly evaluated upper and lower
limits for the individual temperature dependent reaction rates. We found that
most of the uncertainty in the final abundances is caused by uncertainties in
the neutron capture rates, while $\beta$-decay rate uncertainties affect only a
few nuclei near $s$-process branchings. The $s$-process in rotating metal-poor
stars shows quantitatively different uncertainties and key reactions, although
the qualitative characteristics are similar. We confirmed that our results do
not significantly change at different metallicities for fast rotating massive
stars in the very low metallicity regime. We highlight which of the identified
key reactions are realistic candidates for improved measurement by future
experiments.
