  Even though a method to perfectly sign quantum messages has not been known,
the arbitrated quantum signature scheme has been considered as one of good
candidates. However, its forgery problem has been an obstacle to the scheme
being a successful method. In this paper, we consider one situation, which is
slightly different from the forgery problem, that we check whether at least one
quantum message with signature can be forged in a given scheme, although all
the messages cannot be forged. If there exist only a finite number of forgeable
quantum messages in the scheme then the scheme can be secure against the
forgery attack by not sending the forgeable quantum messages, and so our
situation does not directly imply that we check whether the scheme is secure
against the attack. But, if users run a given scheme without any consideration
of forgeable quantum messages then a sender might transmit such forgeable
messages to a receiver, and an attacker can forge the messages if the attacker
knows them in such a case. Thus it is important and necessary to look into
forgeable quantum messages. We here show that there always exists such a
forgeable quantum message-signature pair for every known scheme with quantum
encryption and rotation, and numerically show that any forgeable quantum
message-signature pairs do not exist in an arbitrated quantum signature scheme.
