EDIT: Here's the link to the prodigy spreadsheet, will be filled out as people contribute


I'm doing it this way because these are the stand-out concerns, and thus the most critical issues for achieving any sort of balance with prodigies. The top three dominate all the others for either utility or raw damage output. Also, there are many weaker/useless prodigies... but the recommendations and fixes for them would be more judgement calls/game design questions, so I would like people's input on them. Sadly, probably about half the prodigies are unusable in that they're dominated for whatever they do (Offense/defense/etc) by at least two others. And, of course, while there are stand-out top tier ones, this also limits your reasonable choices.
Overpowered Prodigies
In order, the ones that are the biggest balance concerns are:
Swift Hands
Temporal Form
Mental Tyranny
Swift hands: There are just too many ways that swapping out your entire inventory every turn is overpowered, and bad from an overall balance perspective as it makes balancing items themselves problematic. Swap to max temporal resist when damage smearing starts ticking hard without losing turns, +move items when you need to escape, weapons each turn for special effects, spellpower gear to use an item that depends on it, etc.
I believe making it once per turn would solve any overt abuse, while still leaving it powerful enough to be worth consideration.
Temporal Form: The problems with this prodigy are exacerbated by, but not dependent upon, swift hands. This is by far the highest damage prodigy, for several reasons. First, like mental tyrany, converting all dmg to one type and allowing easier stacking (I'll discuss that with Mental Tyranny). More importantly, when your highest dmg is converted to Temporal plus the boost, you can then swap out just a few pieces of bonus, say, blight dmg gear for +crit or +temporal gear. Telos's top half -> corpathus, Armor -> temporal augmentation robe, trinket -> bladed rift, etc. It just allows ridiculous burst. Also, temporary bonuses aren't lost, such as corrupters temp bonus blight damage on crit, which allow an extra ~50% blight damage -> temporal form, and you just get the boosted damage for all 10 turns of it.
Recommendation for this would be to make your boosted temporal damage reactive to the damage type that it got boosted from. Such that, say, your 50% bonus blight damage buff wears off, lose 50% temporal damage as well, etc. I would also recommend reducing the bonuses somewhat as it already gives you various immunities/spells (they should be buffed, though)/consistent type dmg.
Mental Tyranny: This is by far the second highest damaging prodigy. Both because it allows you to focus on only 1 damage type, and also because it functionally just gives you a strait 40% damage increase. 40% added on to late game crits is way, way more than any of the on-use damage prodigies.
For this, I think that the all 1 type of damage is fairly huge, and really the key/interesting thing. I'd suggest that the bonuses get toned down significantly. They should be weaker than Temporal forms bonuses, certainly, as Temporal is an on use with a timer/cooldown, where Mental Tyranny is a sustain.
Underpowered Prodigies
These were the prodigies for which the consensus was that they definitely need some love.
I Can Carry The World!
Through The Crowd <--- worst two for sure
Superpower
Fungal Blood
Tricks of the Trade
There were many others that are clearly borderline, and would require some boost/change of some sort to make them compelling options. Suggestions and such are on the spreadsheet
