In regards to last week. That was our confirmed cases. This will be speculation based on those cases. Among the three demigods today are Hazel Levesque (Daughter of Pluto), Frank Zhang (Son of Mars and descendant of Poseidon), and Leo Valdez (Son of Hephaestus). Let's begin the analysis.
We'll start with Leo Valdez. His natural abilities include fire manipulation, natural weapons building and technical capabilities (beyond most demigods in some cases), and the ability to protect himself from harm by fire if he concentrates. How this would come into play in Tartarus with the environment itself consisting of a river of fire and various metals that can be crafted into destructive weapons is likely to be incredibly advantageous to the young man. Although, would he actually need to drink the waters of the Phlegethon or would he be capable of healing completely through bathing in it similar to the Styx with Percy? We do know he doesn't have an aversion to being doused with oil and even makes a special mix of crude oil and Tabasco sauce for his automaton dragon Festus, and, according to Annabeth, the Phlegethon is similar in taste to what would likely be this exact mixture. In her words, drinking from the River of Fire was similar to drinking a ghost pepper smoothie. A single ghost pepper alone is enough to burn away taste buds, however a smoothie would likely be at the level of battery acid in terms of acidity levels. In the end, Leo would likely not jump in, yet, after his fight with Gaia, he would likely survive an extended period in the flames. In terms of making weapons, it is likely a lost cause outside of crude weapons fashioned from the beaches of glass or upgraded from fallen enemies. None of which would last, and, knowing Flaming Valdez, he would likely resort to basic tools from his magic tool belt. That is, if it stays on him after the fall or even any entry to the Pit. Although, after his encounter with the dwarves in Bologna, he would likely have developed a way to keep it in almost any situation knowing it was going to be stolen again at some point. In terms of ability enhancement, I don't think we'll see that much outside of an inability to control his flames similar to his usage during his childhood. Although, in Tartarus, it would likely do less damage than in the mortal world due to less structures and, more importantly, a lack of things to burn outside of Damasen's domain. In the end, the only real changes would be an increased building potential which would likely go unused.
Next, we have Hazel Levesque, Daughter of Pluto and Champion of Hecate. She is known for controlling precious metal, having perfect bearings in subterranean areas, and for controlling the Mist itself by having people see what they wish. Although, we do get a more accurate description when she is with her brother in Son of Neptune. According to her, she had one side of Pluto's sphere of control and her brother had the other. Her brother controlled the dead, while, in her own words, she controlled riches. In other words, she may have the ability to control any desired object as long as it a treasure to the person involved. It may also explain her questions' wording throughout Son of Neptune while calling forth various metals. Such as when the fan favorite schist was summoned, she didn't ask what it was alone. She also asked, was it valuable? A question that she seemed to ask about any unknown metal she would summon, almost as if to gauge whether she could send more. Now, being in Tartarus, she would likely be surrounded by many precious metals above her as well as various unknown metals monsters would use in armor. However, her underground senses would be difficult to follow with so many different paths around her as seen when in Rome. The main thing that would likely be increased would be her ability to see what each monster wanted most, likely escape from Tartarus. That would turn her into a gps of some sort for the monsters, and likely cause a lot of trouble for herself and the mortal world. Both by being a greater target for capture, and, once captured, a demigod they would not allow to leave their sight. In the end, she would be in a worse position than her brother.
Last one for today, Frank Zhang. His gifts are a lot more rare. First of all, he's a tactician. He may not have the rage of his siblings, but he can get out of many situations with ingenuity alone. His family's ability is a bit harder to pin down. According to his grandmother, if you know more about what you want to become, it's easier. Although, combat situations are when shapeshifting is done with almost no trouble. What would that mean when he's in a place constantly against him? Would he be able to transform into monsters, as well? Or would transforming place him in greater danger because animals aren't suited for Tartarus? We have seen various monsters force him into his natural state with various abilities. Among these are surprise and chemical based attacks. In the end, I believe just being in Tartarus will slowly push him to being like his fellow children of Mars. Almost to the point of being a lone fighter against an horde of monsters a la Doomguy, although it may be closer to Rambo. Even to the point of being attacked on all fronts, but constantly using guerrilla warfare to keep ahead of each attack and continue fighting until he could escape.
This one is done. Next week I can (hopefully) do Jason Grace, Piper McLean, and maybe two special non-demigods.