I doubt they’re quick and easy to make or else everyone would have one, and if the Ministry is the main source of them you may recall the Ministry was having issues of its own by book 6.
A better example is the Felix whatsit potion that makes you lucky - Slughorn has a whole cauldron full of the stuff at the beginning of book 6, of which Harry only uses a tiny amount. What happened to the rest of it? That stuff would have been mighty handy for dealing with their problems.
Didn’t they drink it all at the end of HBP to protect themselves from the Death Eaters? No reason they couldn’t have brewed more, I guess.
Maybe wizards have the same attitude towards potions as I do in RPGs - I never drink them because I might need them more later.
Wasn’t Hermoine’s time-turner limited to an hour? Was there any suggestion that other time-turners go back even further?
If not, then there’s an answer: you can’t use it unless you get one to the crime scene within an hour.
I think they go back three hours in the end, but there’s no suggestion of a technical limit.
Maybe someone made the only ones. We know that Snape invented some sort of nasty bleeding spell that he apparently didn’t share, IIRC.
Possibly Janus Clocksmith figured out how to create them, made a few, and died before sharing the secret. The Ministry would have been quite interested in gathering them up to prevent mischief.
That potential rarity makes it less likely that Hermione would get one to take a couple of extra classes, of course.
I suppose that would have to be the difference between magic and technology: magic can’t be reverse-engineered. Given a real-world time machine built with modern technology, a team of engineers and physicists could figure out how it worked and build more, even if the inventor was dead and all his notes were gone. Just knowing that something is possible is a huge help when trying to reproduce it yourself. Observing an invention in use gives great hints to someone who knows what to look for as to how it works. Apparently, this isn’t true of magic - it seems to be impossible to take a device like the Time-turner and figure out how to build your own merely by examining it.
There’s also remarkably little mass-production for things other than fairly trivial charms shown. It’s as if only Steve Jobs could make iphones, and once he died the secret of how to make them died with him.
I seem to recall that how far back you go is determined by the number of times you flip it, so if you attach it to, say, a power drill, you should be good to go back to the Jurassic.
If they used a power drill you’d end up with even more plot holes.
It is, and so she flips it three times to go back three hours. As noted, there’s no issue there, nor do we have any reason to believe that all Time-Turners are calibrated for one hour increments. Indeed, it stands to reason that the Ministry would issue Hermione a small one; it’s all she needs for her lessons.
And besides, weren’t the authorities on the crime scene inside of an hour? Black was still cackling when they got there, wasn’t he?
With respect to the Priori Incantatem* spell, all it proves is the last spells (or maybe last few spells) cast by a particular wand. It gives no clue as to who cast the spell, as Winky learned in Goblet of Fire. Memories can be altered by magical means. All Veritaserum demonstrates is what the subject believes to be true, and as noted, memories can be altered by magical means. And maybe most damning, the MoM in Britain is a little fascist. We really see few or no trials in which anyone really tries to ascertain guilt or innocence; rather, we see a bunch of show trials in which the whole court has already made up their minds about the outcome. If they didn’t already know if the person was guilty or innocent (often for political reasons) the trial didn’t happen.
*It’s bad enough that I remember that off the top of my head. I refuse to actually look it up.
It doesn’t appear to be up anymore, but JK Rowling’s official website used to have a section where she answered questions from fans. One was about the use of Veritaserum, etc., in trials, and she said that these methods only carried about as much weight as polygraph testing does in the real world. A sufficiently prepared, skilled wizard could manage to keep his secrets. We know from the books that it’s possible to tamper with memories viewed using the Pensieve (Prof. Slughorn did it), that it’s possible to shield one’s thoughts from mind-readers (Snape was very good at this), and that it’s even possible to magically wipe memories (Gilderoy Lockheart did it repeatedly). IIRC it was specifically mentioned that Veritaserum wouldn’t force someone to reveal a secret that was protected by the Fidelius charm, and there may have been other magical ways around it as well. There were certainly non-magical ways – in Order of the Phoenix Snape administers a different potion to a student and just tells Umbridge it was Veritaserum.
But more importantly:
This. We’ve had threads here before asking why the wizard justice system doesn’t use Veritaserum, etc., to establish the truth in trial, but even if these methods were more effective it’s clear that the wizard government is not always interested in the truth. This isn’t a plot hole, it’s a significant part of the plot across the series that the government is often at best incompetent. Many officials are corrupt, prejudiced, or basically fascists. I don’t remember the specific details of the trial scenes that well, but unless it’s explicitly stated in the books I don’t think we can assume that defendants are presumed innocent until proven guilty, have the right to an attorney, etc. This is a government that tortured convicts, after all. Did Sirius Black even have a trial, or was he just thrown into Azkaban?
Even with a fair legal system it probably would have been pretty easy to convict him, though. There would have been no need for the prosecution to use Veritaserum to get a confession. If Black offered to take the potion to add weight to his testimony, it would be assumed that he was only making the offer because he was confident he could beat it. Using a Time Turner seems a lot more trouble than it would be worth, as 1) there were already witnesses to Peter Pettigrew’s “murder” and 2) going back farther to verify who betrayed the Potters would mean going back to a time before Voldemort was defeated. Although the third book indicates that time travel cannot be used to alter the past, this doesn’t seem like the best scenario for putting that to the test. And there’d be no paradox if the time traveler would up being killed or cursed by Voldemort or the Death Eaters. Really I can’t see even the most gung-ho defense attorney feeling it was worth the risk, even if s/he’d been convinced of Black’s innocence…and remember that Lupin, his closest living friend, didn’t believe he was innocent at the time.
Actually no, there is no due process. Hagrid was taken directly to jail without a trial and it was inferred that Stan Shunpike was railroaded into prison. And I left off another plot hole. They have the ability to read minds. And as far as the truth serum is concerned, even He Who Must Not Be Named couldn’t fight 3 drops of the drug.
The books are so full of plot holes I could right another book on them. That’s just a function of using magic and time travel in a story. You can’t escape them. It’s best to enjoy them for what they are and not let logic get in the way of a bit of fantasy.
I enjoyed reading the stories as well as the movies because they allowed me to study screenplays. It was fun to see how a story can be truncated and rearranged with destroying it. I think the screenplays were brilliant.
One of the other Death Eaters used that same spell during the battle at the Ministry. Harry just didn’t know that it was the same spell, when he used it.
By the time of the last movie, Hermione’s charms were considerable! 
The underlined appears to be an assumption. Do we really have that demonstrated in the text? The way I see it, you typically employ an owl who already knows where the person is, or I presume the owl flies to some distribution center to find out where that person is. Hedwig is described as flying off for long periods of time and coming back exhausted, so it does not appear that the owls teleport or otherwise magically transport, but rather they actually fly wherever they go.
If the owl didn’t already know Sirius Black’s location, it wouldn’t be able to find him. Either Hedwig or Harry know some trick of how to get messages to Sirius. Maybe it’s a “go to location X, read note, proceed to location Y” kinda thing.
That would be sensible, likely another reason why Sirius instructed Harry not to use Hedwig. “Dammit, why can’t we be as smart as these readers! We should have thought of that first!”
Yep.
You see, it just takes someone else to outthink the problem. Maybe the Ministry did try tracking owls, but Sirius had already outwitted them.
But, she’s Hermione, the best witch of her age! Of course she deserves a time turner merely to take extra classes, and totally not to use for any other reason. 
Bravo, sir. You just combined Chaum’s Mix Network with RFC 1149.
One thing to keep in mind is that the MoM is not really that effective as a government body. They operate at about the level of a municipal government (and, indeed, all the “citizens” they govern are probably only equal to the population of a large town/small city). When Voldemort was in power, he wasn’t a crime lord–he was a warlord. If the cops have a warrant on someone like Al Capone and knew where he was, there is no doubting the outcome–Capone loses. Not so with the MoM and Voldemort. It wasn’t a matter of, “if we can FIND Voldemort, we can stop him!” The whole of the MoM was not powerful enough to stop him. He died by mischance. You would expect an effective government to have an effective law enforcement/military. The MoM does not.
As a result, many of their other law enforcement/governance failures are not hard to understand.