  We revisit recent works by Don, Fehr, Majenz and Schaffner and by Liu and
Zhandry on the security of the Fiat-Shamir transformation of $\Sigma$-protocols
in the quantum random oracle model (QROM). Two natural questions that arise in
this context are: (1) whether the results extend to the Fiat-Shamir
transformation of multi-round interactive proofs, and (2) whether Don et al.'s
$O(q^2)$ loss in security is optimal.
  Firstly, we answer question (1) in the affirmative. As a byproduct of solving
a technical difficulty in proving this result, we slightly improve the result
of Don et al., equipping it with a cleaner bound and an even simpler proof. We
apply our result to digital signature schemes showing that it can be used to
prove strong security for schemes like MQDSS in the QROM. As another
application we prove QROM-security of a non-interactive OR proof by Liu, Wei
and Wong.
  As for question (2), we show via a Grover-search based attack that Don et
al.'s quadratic security loss for the Fiat-Shamir transformation of
$\Sigma$-protocols is optimal up to a small constant factor. This extends to
our new multi-round result, proving it tight up to a factor that depends on the
number of rounds only, i.e. is constant for any constant-round interactive
proof.
