Just before Voyager transmits The Doctor to the Alpha Quadrant, he steps upon a platform in Astrometrics, still wearing his mobile emitter. It can still be seen on his left arm as he disappears as he is transmitted. Since the mobile emitter is made of real matter, it should've stayed behind and clattered onto the Astrometrics' platform when the Doctor vanished.
Just after the battle with the Romulans ends, a Starfleet security team beams aboard the Prometheus. The two Doctors never lowered the Prometheus's shields. No one should be able to beam aboard with shields raised.
Seven states the ship is at the outer edge of the alpha quadrant, near the boundary of the delta quadrant, on a deep space mission, yet her map shows the ship to be at the outer reaches of the relay network which extends well into the alpha quadrant, almost to its center.
The main viewscreen of the Prometheus twice shows a stationary starfield despite the fact that the ship is traveling at high warp.
The Prometheus is said to be cruising at warp 9.9 and that nothing in Starfleet can catch them. Minutes earlier, a Nebula class ship, with a maximum cruising speed of warp 9.5, was able to catch up to and attack the Prometheus. No mention is made of when the ship was supposedly travelling slower.
When B'Elanna enters and exits Sickbay to fetch the doctor, actress Roxann Dawson 's pregnancy is apparent. Incorrectly regarded as a goof: What you're seeing is the loose knee-length smock the character is wearing (unique to this episode) fluttering around as she rapidly enters and exits. Presumably this dress was worn to disguise the actress' condition.
The amount of time that passes between the time Seven says "35 seconds" and "10 seconds" is only 8 seconds. Incorrectly regarded as a goof: The scene is edited and not shown in real time.
When the Doctor first attempts to pilot the Prometheus, he sits at the starboard forward bridge console, which is the traditional helm position on a Starfleet vessel. Later, when the Romulan Warbirds are advancing on the Prometheus, the Doctor is attempting to pilot the ship from the port console, with the EMH mk 2 seated at the starboard console attempting other tasks. Incorrectly regarded as goof: Since this ship is acknowledged to be a new design, the console positions may not be static, and helm control could be available as a standard at multiple consoles; even other familiar Starfleet vessels have worked the helm from other stations on the bridge and even elsewhere on the ship such as engineering.
When Seven of Nine tells everyone that they've got "90 seconds" before the ship moves out of range, the amount of time that passes between that statement and the next one telling them they have "35 seconds" is only about 20 seconds. Incorrectly regarded as a goof: The scene is edited and not shown in real time.
After Harry and Tom bring the EMH replacement online, the new Doctor can be seen visibly wobbling in place when he should not yet be mobile. His head also moves when struck by Lt. Paris even though his interactivity has yet to be installed.
Near the beginning, it is clear that Seven of Nine has pierced ears. This is not something that the Borg would have done to her, nor is it likely that she would adopt this earth custom as part of her reinstating her individuality.
After the Doctor and the EMH-2 are taking care of the Romulan in Sick Bay on the Prometheus and the scene cuts back to Voyager, the camera closes in on Paris at the helm. Then the scene changes to Sick Bay on Voyager and Paris can be seen walking in saying "The Doctor is in." There is no way he would have had time to leave the bridge and get down to Deck 5 so quickly.
The Prometheus is an advanced ship, superior to the other ships in Starfleet such as Galaxy class and possibly even Sovereign class ships like the Enterprise E. Two issues with this. First, you never see any of these ships when fighting the Dominion, and it was explained this was one main reason for building this ship. Second, the technology available on the Prometheus, such as ablative armor and regenerative shields make much sense to incorporate into other Starfleet vessels to fight against the Dominion, yet Starfleet seems to ignore that by allowing their older vessels to age and retain their older and outdated technology.
At around 10' the Romulan commander says "You'd be surprised how stubborn Humans can be", but on the word "humans" his lips are saying something else.
Uploading The Doctor's software to a different ship would not delete the original.
The Doctor informs the injured crewman that he has "severe phaser burns." Romulans do not use phasers, they use disruptors. It has previously been established that phaser and disruptor blasts are distinct and easily differentiated.
With time running out to send a message through the relay network, Captain Janeway orders Torres to go to sickbay, get the Doctor, and meet her in the astrometrics lab. Torres walks to sickbay, downloads the Doctor to his mobile emitter, then walks to astrometrics. Since time was of the essence, it would have been far more efficient for Janeway or Torres to simply call the Doctor and ask him to meet them in astrometrics.
Shortly after arriving in the Prometheus' sickbay, the Doctor discovers burns on the wall and the bodies of two crew members in operations-gold uniforms. He practically has to step over the female officer to get to the burned wall, yet somehow doesn't notice her until after examining the scorch marks.
After a single volley of fire from the Romulan Warbird, the EMH-2 states that Prometheus' shields are brought down to 20%. The Prometheus is supposed to be an advanced prototype with far greater defensive capabilities than most starships. Galaxy class ships, for example, have been seen taking far greater amounts of fire from Romulan warbirds without suffering nearly as much damage. Furthermore, If a single volley of disruptor fire takes their shields down to 20%, then a second shot should destroy them. Yet, additional disruptor fire does not destroy the Prometheus. Also, the shields cannot still be damaged after the initial Romulan takeover of the ship, as the technical readouts stated that the Prometheus has self-regenerating shields.
The Prometheus is said to be traveling towards Romulan space at warp 9.9. The EMH mk 2 declares that the ship was designed to go faster than any other vessel in the fleet and that no Starfleet ship could catch them. The Intrepid class, of which Voyager is a part, has been in service for at least 4 years by this point and has a maximum cruising speed of warp 9.975. The Sovereign class, like the Enterprise E, can reach speeds of warp 9.995. According to several other Star Trek episodes, most notably VOY: "Threshhold", nothing can travel at warp 10 or above. Faster than 9.995 but less than warp 10, is such a negligible amount of speed difference, it would hardly be much of a boast to say that the Prometheus can travel faster than any ship.