During the first fight in the arena, Flavius is struck twice, vigorously, by a bull whip, and yet on subsequent shots from Flavius' rear, there's no evidence of any kind of wound whatsoever on his back, which there surely would have been.
Max Kleven is billed as "Maximus", but his character is referred to as "Achilles" in the arena fight.
Near the end of the program, the viewer gets a look at the planet from space, and we can see two moons off in the distance, with the sun shining from off screen right. In a subsequent shot, from the bridge of the Enterprise, we see the same thing, except that the sun is shining from the left onto the two moons, but still from the same direction as before onto the planet below.
Near the end when Kirk, Spock and McCoy are beaming out of the jail cell the guards start shooting machines guns at them using a sweeping motion from side to side yet the patterns on the wall show three tight groups.
Drusilla tells Kirk, "for this evening, I was told I am your slave." Later, Marcus awakens Kirk and says "I don't wonder you slept through the afternoon."
When viewing the initial television broadcast with mentions on slavery and gladiators, Kirk comments "What are we seeing, 20th century Rome?" Obviously it would be more like 1st century Rome, but Kirk is clearly referring to the Roman Empire brought into the 20th century, not the city of Rome in the 20th century.
As Flavius takes Kirk, Spock, and McCoy towards the city and hiding behind the trees, red spots can be seen on various places of the trees. The small charges were set off when guards fired their machine guns at the group.
During the jail scene with Spock and McCoy together in the cell, the joint on Spock's prosthetic pointed left ear can clearly be seen peeling.
When Spock is straining at the jail cell door his face flushes red. As Spock is Vulcan and has a copper based hemoglobin his face would flush green rather than red.
When Kirk uses the machine gun to free McCoy and Spock, his aim at the cell door lock is way off, and you can easily see there are no bullets hitting the lock, yet it opens.
When Kirk goes to free Spock and McCoy from the jail, he uses the machine gun to shoot out the lock. But when he opens the gate, the lock is undamaged.
As Claudius Marcus drinks a final toast to Kirk before he sent off to die, the prop number can be seen at the bottom of the goblet. It happens about 44 minutes into the episode.
During Kirk's log entry about being in a modern-day Rome, we are taken on a "tour" of the city's landmarks, including the main library at the Massachusetts Institute of Technology (plainly readable on the facade) and the California Palace of the Legion of Honor, in San Francisco (as evidenced by the French - not Latin - inscription "Honneur et Patrie" on the building).
Flavius is guiding Kirk, Spock, and McCoy into the city during daylight. They hide behind some trees and Flavius says "we will hide here until dark." The four are poorly hidden by the trees and talking, so, quite naturally, they get caught. They could have stayed hidden in their cave until dark, then went into the city, and would have been far less likely to have been caught.
The entire premise that another world would develop a Roman Empire, complete with Roman names and titles, but also speaking English to the extent of having the same son/sun homonym confusion, is totally ludicrous.
Episode is based on a Parallel Earth Development theory, where Rome does not fall. Spock expresses surprise when their first encounter with a local, Flavius speaks English. The Universal translator always translates to English.
Dr. McCoy says ancient Romans were not sun worshipers. Several Roman religions included solar deities, including Apollo, Mithra, and Sol Invictus.
Spock says that 6 million died in World War I and 11 million in WW II. These figures aren't even close. 9 million combatants and 7 million civilians died in the first world war and well over 50 million total in the second; some historians believe possibly as many as 85 million. Around 11 million died in the Holocaust alone, if you include non-Jewish victims of Nazi mass murders.
Kirk, Spock, and McCoy discuss the matter of a parallel Roman Empire in the 20th century, and find it logical that such a civilization would speak English. Shouldn't it speak Latin, or even a modified derivate such as Italian?
Kirk, Spock, and McCoy knowingly beam down to a 20th century planet to covertly look for survivors from the Beagle. They don't disguise themselves in 20th century clothing, they beam down in their very out of place uniforms, and very quickly get caught.
The police officer who captures Kirk and company says it has been "too long" since he'd seen barbarians die in the arena, yet, the newscast Uhura picked up earlier stated that one named William B. Harrison had died the day before. There are two possible explanations for the officer's remark. Perhaps he was on duty the day before and didn't see Harrison die. Or, perhaps even one day since the seeing a barbarian die is "too long" for him.