It would depend a great deal on the context. Who is speaking and where? But since Broadway is internationally famous and prestigious, the musical's promoters, for example, would no doubt refer to it as a Broadway musical.
Musicals that have played on Broadway (or London's West End, I presume) are assumed to have a certain level of quality, style, sophistication, etc. or they wouldn't have made it through the process of making it to Broadway. This continues to be true when they are no longer playing on Broadway.
Harvard graduates continue to be just as smart as they had to be in order to be accepted to study at Harvard.