The introduction of Star Trek mirror universe. As with the mirror universe, Picard's Confederation is a brutal, militaristic version of the egalitarian Federation that audiences know and love. However, the Confederation seems even more sadistic than the mirror universe, staging public executions of their enemies to entertain the masses. By comparison, the Terran Empire appear humane, allowing the two enemies to confront each other in a battle to the death, something which is a tenet of their power structure.
The Confederation was created when Q (John de Lancie), with the help of Dr. Adam Soong (Brent Spiner) sabotaged a historic space flight that opened humanity's eyes to the wider universe. The result was that humanity became increasingly insular and protected their own interests instead of working with other civilizations to improve the galaxy. The Terran Empire's domination of the galaxy had a similar origin when Zefram Cochrane killed the Vulcan first party and used their advanced technology to begin conquering the galaxy. Despite the Terran Empire's strength, however, the Confederation's casual attitude to brutality would give them the edge in a battle for supremacy.
Picard’s Confederation vs. The Mirror Universe’s Terran Empire
Based on their respective Star Trek timelines alone, it appears that Picard's Confederation would defeat the Terran Empire. In the Confederation reality, the regime has a staying power that the Terran Empire lacks in the mirror universe. Not only has the Confederation been dominating the galaxy for longer than the Terran Empire, but it's far more fearsome, with Jean-Luc Picard and his Enterprise held up as great warriors. In Picard season 2, episode 2, a painting on the wall of Jean-Luc's study portrays the Enterprise destroying a Borg cube.
It's later revealed in the same episode that the Confederation wiped out the Collective and is keeping the Borg Queen as a prisoner. It's her execution that is staged to the public as part of the annual Eradication Day celebrations overseen by Confederation President Annika Hansen (Jeri Ryan). If the Confederation has the technology and the drive to destroy the Borg Collective, then the Terran Empire don't really stand a chance, especially as they were already toppled by the mirror universe's Klingon and Cardassian Alliance, following radical reforms by Emperor Spock.
Picard’s Confederation vs. The Mirror Universe’s Klingon/Cardassian Alliance
From the gruesome artifacts in Picard's study, it's clear that the Confederation could easily see off a potential Klingon and Cardassian alliance. In Picard season 2, episode 2, "Penance", Q takes Jean-Luc on a tour of his new home. Rather than the idyllic vineyard from Picard season 1, the château has become a macabre museum that celebrates the military exploits of General Jean-Luc Picard. Among the exhibits are the skulls of the Cardassian Gul Dukat and Klingon General Martok, prominent military figures, which confirms that a prospective Klingon and Cardassian alliance would be no match for the brutality of the Confederation.
In DS9, it's stated that Emperor Spock's radical disarmament program weakened the Terran Empire, a weakness that was exploited by the Klingon and Cardassian alliance to turn the tables. From what is show in "Penance", disarmament and reform are clearly not on the mind of President Hansen, and so the Confederation's dominance over the dark Star Trek: Picard timeline continues unabated. Therefore, the Confederation would defeat the Terran Empire in a battle of the darkest Star Trek timelines.