InThe Crucible, the witch trial hysteria is fueled by people's grudges and personal rivalries. Many characters use the witchcraft trials for revenge and their own personal gain. For example, Abigail Williams is able to accuse Elizabeth Proctor in hopes of eliminating Elizabeth as a rival for John Proctor's affection. Thomas and Ann Putnam, jealous of Rebecca Nurse's large family, use the witch trial hysteria to have Rebecca charged with the supernatural murder of their dead babies. Martha Corey is accused of witchcraft by a man upset with her for not giving him his money back for a pig he bought from her but didn't take care of properly. The backdrop of the witch trials created an opportunity for people to act on their long-held grudges. Unfortunately, many of them did act on this and added fuel to the fire thus elongating the witchcraft trials.
It was quite easy, in the real witch trial era, in the play, and in the 1950s McCarthy era the play was mocking, to get rid of rivals or benefit personally from publicly accusing someone of witchcraft. Rumors were enough, so the accusation didn't even have to be direct.
In the play itself there are numerous examples of rivalries that got swept into the witchcraft hysteria. There's Mr. Putnam, who is always in land disputes with his neighbors, including Proctor and Giles Corey. So he would benefit from their being arrested or hung. There is Abigail and her rivalry with Proctor's wife, as she is jealous of Elizabeth's being married to John. In addition, there is the rivalry between Proctor and Reverend Parris, as Parris tries to save his adopted daughter and his own good name in the town, and is willing to see John go down in pursuit of those goals.
So rivalries accelerated and intensified during the witchcraft hysteria, in real life and in The Crucible.