Would the Diaspora be a time of persecution in Judaism?

The Diaspora is the state of exile of the Jewish people. There isn't necessarily a time called "the Diaspora"; however, in most of the state of exile, Jews have been persecuted heavily. Even today, Jews are persecuted in several countries, though not nearly to the extent that it used to be. So yes, Jews have historically been persecuted in the Diaspora.