  Recently, the concept on `forgeable quantum messages' in arbitrated quantum
signature schemes was introduced by T. Kim et al. [Phys. Scr., 90, 025101
(2015)], and it has been shown that there always exists such a forgeable
quantum message for every known arbitrated quantum signature scheme with four
quantum encryption operators and the specific two rotation operators. We first
extend the result to the case of any two unitary rotation operators, and then
consider the forgeable quantum messages in the schemes with four quantum
encryption operators and three or more rotation operators. We here present a
necessary and sufficient condition for existence of a forgeable quantum
message, and moreover, by employing the condition, show that there exists an
arbitrated quantum signature scheme which contains no forgeable quantum
message-signature pairs.
