How does the quote, "Power tends to corrupt, and absolute power corrupts absolutely," apply to Julius Caesar?

In Julius Caesar, the plot of the play is set into motion by the fear of Cassius, Brutus, and the other conspirators that power will corrupt Caesar, and he will become a dictator. He is wildly popular among the people and has just returned in triumph after defeating Pompey's sons. Brutus and Cassius and the others decide to murder Caesar while they still can—before he has amassed too much power. Implicit in their act is the fear that absolute power corrupts absolutely.
As is so often the case, their motives are mixed: people like Brutus genuinely do want to save Rome from tyranny, but they are also are jealous of Caesar and fearful for themselves and their own power.
Brutus, Cassius, and the other conspirators do kill Caesar, but they safeguard neither themselves nor Rome by this act. They set Mark Antony and Octavius against them and spark a civil war that is both ruinous for Rome and for themselves personally.
The conspirators consider killing Mark Antony, too, but decide not to, foolishly writing him off as a partying playboy. That is a huge mistake.
As Mark Antony amasses power against them as a military commander in the civil war, he shows that power corrupts. He is genuinely grieved and deeply angered over Caesar's death, and the warmth of his feeling floods his being and helps him make a convincing speech that sets the crowd against Brutus. Antony jeers at Brutus and the other assassins as "honorable men." Yet, he is turned hard by power. For example, he cold-bloodedly decides to trade the death of his own nephew for that of Lepidus's brother, seeming not to care about his nephew's life. Antony also coldly treats Lepidus as nothing but a weak tool to be used and discarded, saying he is

a slight unmeritable man, / Meet to be sent on errands

Nobody in the play, however, gains absolute power, so nobody, not even Antony, is corrupted absolutely. Nevertheless, a theme Shakespeare explores is that of the limits of power: how much power is too much for one person to hold? To what lengths can people go to limit one man's power? Why does getting power, as in the case of Mark Antony, so often lead to cold-blooded, rather than humane, behavior?
In Julius Caesar, Shakespeare both suggests that absolute power is corrupting and, at the same time, argues that illegal means to block power, such as assassination, unleash more problems than they solve.