During its original run and well into syndication, I couldn't stand Quantum Leap — the "rules" governing Sam's time travel made no sense to me. "He's bouncing around from time period to time period, and he's stuck in each life until he makes things... nice??" One weekend, the Sci Fi Channel ran a marathon of the final episodes of various shows, and I happened to catch the finale of QL, which reveals that the controlling impulse behind Sam's "leaping" is his own subconscious drive to make the world a better place, one life at a time — and I then proceeded to catch the show five days a week for the next few months. — Your Obedient Serpent
The other thing that I realized (then Googled and realized to my disappointment I wasn't the first) was that, especially clear during the last season, he was realigning the "Ziggy Version" of history to match our version and that after those "major" events in modern history were lined up, he was done except for one last thing (having Beth wait for Al) which completed the alignment and therefore returned to his "Ziggy History" timeline and disappeared from our timeline (thus ending the series) — Gaidin BDJ
In "Star Light, Star Bright," Sam is hypnotized, and on tape, gives away the secret access code name and numbers regarding top secret information about Project Quantum Leap. An episode later, we meet an evil leaper and hologram. Clearly, someone else found the tape, accessed PQL's files, and set up their own project for personal gain!
And thanks to the events of "Shock Theater" and "The Leap Back," one of the Project's handlinks is left stranded in the 1940s and unaccounted for. It won't work for another 50 years, but it's completely intact and undamaged. Between that tape and a piece of technology that could be reverse-engineered...
Speaking of "Shock Theater," which takes place in 1954, a bunch of doctors are seen analyzing Sam's multiple personalities, which are all of people that Sam encountered in years after 1954. Maybe someone uncovered a doctor's notes and realized that the mental patient wasn't just making lucky guesses, but was actually a Leaper at the time...
There is a LOT of information that Sam reveals to others about PQL. Sam reveals a bunch of info about himself, PQL, Al, Ziggy, and Gushie to Carol and Becky Pruitt in "Killin' Time," which was two leaps before Sam first encountered Alia.note Becky even tells that episode's sheriff that Leon Styles (the leapee) is actually Sam Beckett.
That explains why the evil leaper organization does everything exactly the same with different sound effects; hologram technology, handlink, etc. Plus, Zoey was aware of just how long the window was before a successful retrieval could be performed, mentions that Lothos has some control over Alia's leaps, and outright states that Sam uses a random leaping sequence — which means the evil organization may have improved on PQL's math.
Throughout the series, Sam displays a reluctance to engage in any sort of relations with women, a fact that Al laments constantly. Near the end of the show's run, it is revealed that Sam is married, and suddenly his hesitation can be interpreted as a subconscious reaction to the possibility of cheating on his wife.
Donna states that she does understand, knowing that he can't remember her and is doing it to improve the lives of those involved.
There's also the fact that when he did sleep with someone in "Trilogy", he fathered a child.
Conversely, it can also gives a more unfortunate implication to the times when he does it anyway, or. . .the times when he's leaped into someone already involved in a relationship—the black medical student, for example—he's slipped up and referred to the woman as being in love with him, rather than his leapee. It's possible that his subconscious still remembers his wife and is trying to recreate what he had with her.
In "The Leap Back," when Al drops the letter with the override code (and the $100) to the law firm, with instructions to deliver it on a specific date in the future (a la Back to the Future), the imaging chamber door immediately. This is because, as soon as Al dropped it in, there was no way for him to retrieve it. Hence, the "outside" time was when it was delivered, according to Gushie, that very day.
In The Boogieman, here's an unacknowledged, in-show visual clue: the fake Al wears a simple suit of blue and white with a silver pin on his lapel rather than his uniform or an outfit consisting of bold, clashing colours and/or garish patterns. In fact, he could have visually fit in during the time period Sam landed in.
In "Trilogy Part II" Sam gets hit in the head hard enough to knock him unconscious for several minutes. This trauma also leaves a bloody wound. This is not the only time he is injured shortly before finishing his leap. How is it that his injuries are never there at the start of his next leap? Remember that in the pilot episode Al said it would take several days or weeks for a new person to show up in the waiting room. Sam's body is healing during the downtime between leaps!
In the last episode, we learn that Sam is controlling his leaps. So why, in Last Dance Before Execution, did he wait until the very last moment before leaping? Because he didn't want Jesus to have a moment of realisation and horror before he died. Sam stayed as long as possible to spare him that.
In "A Single Drop of Rain" it rains at the end. Considering the way time travel works in the show, if Sam's experiment were going to work, Al would already have been able to observe that fact as a historical event (but he couldn't). Given nothing Sam did on Earth could have changed whether it was going to rain, the eventual rain must have been the result of his appeal to God.
In the last episode, we learn Sam never returns home. In retrospect, over the course of the entire series, it's kind of obvious. Sam has Chronic Hero Syndrome, played utterly straight. It doesn't matter how tired he gets or how much he misses everyone. He knows, always, that at the end of this leap is another person who needs help.
Anyone else find it odd that a Cunning Linguist like Sam didn't know sign language? "Seven modern languages and five dead ones", but not ASL? He also tells AL, "I've never been around a deaf person before". That is highly unlikely, especially considering that an MD is one of the many doctorates he has.
Occasionally, history changes to avoid a disastrous marriage. This was even the point of at least two episodes. The problem? Some of the marriages resulted in children but now, those children will never have a chance to live. Sam and Al are essentially murderers and never show any remorse about it. In one episode Al even gleefully proclaims something along the lines of "Now she won't end up having a couple of screwballs!" Suddenly, Sam and Al don't seem like such nice guys.
It's not murder if you prevent someone from existing but, yes, that is pretty horrifying. There are presumably new kids who exist who live better and happier lives than the old timeline's but saying that the happiness of the parents is worth more than the existence of the children...It's a problem.
I always thought of it like The Inquisitor from Red Dwarf. Instead of children being born who would do nothing with their lives, we get children who never had a chance to exist being born, children who might contribute something to society. And considering that Ziggy never reported that their children did anything (as the OP pointed out, they were a couple of screwballs), then we must assume that this was the case in the other occasions when Sam broke up a marriage.
Plus given the existence of a seemingly benevolent God in this setting, chances are that while the kids as we know them were Ret Gone, they exist in some sense as different kids.
It's been accepted by fans for years that Sam's changes to history were slowly turning his timeline into ours. The problem? His timeline was far more advanced than ours, not even counting time travel. His efforts have resulted in progress slowing down!
It's said several times that Sam can leap anywhere in his own lifetime - but we never see him Leap past his original Leap. Which - to me at least - means that even though Sam's body is leaping around in time, he effectively died at that moment, the events of "The Leap Back" only happening post-Leap in his timeline.
When Sam leaps into Al's younger self, hologram Al takes a ride in a jet fighter before coming to Sam. When he checks the date and time, he realizes that his girlfriend is about to die in a car accident. He teleports to the site of the crash and tries to warn her, but she can't see him (due to not being Sam) and crashes. This was probably Al's worst nightmare. First of all, his younger self only heard about the accident while older Al personally witnessed it. Second, Al tried to warn her about the crash, but failed due to her not being able to see him. There's little worse than watching someone you love die and not being able to do anything about it. This is on top of him not finding out in time due to taking the ride in the jet fighter.
It's frequently demonstrated that children, animals, the mentally ill, and people with brainwaves similar to the leaper's can see holograms. If that's the case, then what kinds of horrible things has Zoey done to these kinds of individuals?
Sam never returned home. Imagine spending the rest of your life in someone else's body other than your own thus robbing every ounce of your own identity.
The "Swiss cheese" memory effect as it pertains to Alia. Sam is able to uncover some sort of hidden morality in her (as opposed to Zoey, who seems to be very much a psychopath), so what kind of person was Alia before the evil organization got hold of her, and how long did they manipulate her amnesiac mind before Sam came along?