The phrase "big shot" means someone who is important, influential or powerful and often uses their position to assert authority. However, there are numerous synonyms that can be used to describe such a person. These include heavy hitter, bigwig, VIP, muckety-muck, top dog, power player, high roller, magnate, tycoon, and big cheese. Each of these terms conveys the idea of someone who is in a leadership role, has significant clout, and commands respect. Using these synonyms can help to provide a more nuanced description of a person's status and influence.