The problem with lay descriptions of spooky action at a distance is they usually describe it in a way that doesn't make at all clear why you'd think there's any action at a distance at all. The classic example of entanglement is you take two particles, and have them interact with each other in such a way that they'll be spinning in opposite directions. Then, without observing which way either is spinning, you separate them over a long distance. At that point, you look at one of the particles, and at almost exactly the same time, somebody else looks at the other particle. You'll both always see them spinning in opposite directions, even if you make your observations so close together that they'd have to be sending a signal between them faster than the speed of light to communicate.
Well, so what? You could do exactly the same thing by separating a pair of shoes. If one person sees a left shoe, the other person will always see a right shoe. There extra ingredient in quantum mechanics is that the particle's spin is undetermined until somebody observes it. The obvious answer to which is: Why would I believe that? Just because you don't know which way it's spinning doesn't mean that there's not some predetermined answer before somebody observes it. And at that point the answer generally comes back: "No, it's not that, trust me lol."
To go beyond that you need to explain Bell's theorem, which is difficult to describe in a layman friendly way. The best explanation I've seen involved lights and boxes- there's a version at https://simple.wikipedia.org/wiki/Bell%27s_theorem but it's not as clear as the one I remember (which unfortunately I can't find)
Just because you don’t know which way it’s spinning doesn’t mean that there’s not some predetermined answer before somebody observes it.
I think part of why it's significant regardless is that identical Fermions (ex: electrons) can't exist in the exact same state as one another, as in you can't have 2 electrons occupy the same atomic orbital subshell with the exact same spin. If the electrons are going to be existing with one another, then they have to have opposite spins, otherwise their wavefunctions would equal zero, meaning they wouldn't exist. But for two electrons that aren't in the same state, then it doesn't matter, they can have either spin state regardless of what the other one is doing.
So by separating two entangled electrons apart, then it should follow that their spin states should be completely arbitrary relative to one another. So you should have states where both are measured to be in spin up or spin down states as well as in opposite spin states. But they're only ever measured to be in opposite states, as if they're still obeying the rules that identical fermions must follow when they try and occupy the same state, despite being far enough away from each other that those rules shouldn't apply.
(Also, I'm just an undergrad and we really haven't covered entanglement, so this is just my rough, and quite possibly wrong, interpretation of why it's important.)
TL:DR: I guess the significance is that while the act of entangling two electrons will necessarily cause them to adopt spins opposite from one another, once you separate them out the relationship that caused them to adopt those opposite spins shouldn't apply anymore. But measurements show that they continue to obey that relationship right up until they're measured.
Edit: Made sure to specify that the rules for Fermions only apply when they're identical/indistinguishable from one another.
Heh, to be honest I can't remember a lot of that! But you're right, Bell's Theorem isn't the only expalantion for why entanglement is "spooky", since Einstein said that long before the theorem even existed. But I think it's the simplest approach for explaining to a lay audience without a lot of extra ideas or "you just gotta believe me on this bit." And I think it's what definitively killed local hidden variable theories.
The problem with lay descriptions of spooky action at a distance is they usually describe it in a way that doesn't make at all clear why you'd think there's any action at a distance at all. The classic example of entanglement is you take two particles, and have them interact with each other in such a way that they'll be spinning in opposite directions. Then, without observing which way either is spinning, you separate them over a long distance. At that point, you look at one of the particles, and at almost exactly the same time, somebody else looks at the other particle. You'll both always see them spinning in opposite directions, even if you make your observations so close together that they'd have to be sending a signal between them faster than the speed of light to communicate.
Well, so what? You could do exactly the same thing by separating a pair of shoes. If one person sees a left shoe, the other person will always see a right shoe. There extra ingredient in quantum mechanics is that the particle's spin is undetermined until somebody observes it. The obvious answer to which is: Why would I believe that? Just because you don't know which way it's spinning doesn't mean that there's not some predetermined answer before somebody observes it. And at that point the answer generally comes back: "No, it's not that, trust me lol."
To go beyond that you need to explain Bell's theorem, which is difficult to describe in a layman friendly way. The best explanation I've seen involved lights and boxes- there's a version at https://simple.wikipedia.org/wiki/Bell%27s_theorem but it's not as clear as the one I remember (which unfortunately I can't find)
I think part of why it's significant regardless is that identical Fermions (ex: electrons) can't exist in the exact same state as one another, as in you can't have 2 electrons occupy the same atomic orbital subshell with the exact same spin. If the electrons are going to be existing with one another, then they have to have opposite spins, otherwise their wavefunctions would equal zero, meaning they wouldn't exist. But for two electrons that aren't in the same state, then it doesn't matter, they can have either spin state regardless of what the other one is doing.
So by separating two entangled electrons apart, then it should follow that their spin states should be completely arbitrary relative to one another. So you should have states where both are measured to be in spin up or spin down states as well as in opposite spin states. But they're only ever measured to be in opposite states, as if they're still obeying the rules that identical fermions must follow when they try and occupy the same state, despite being far enough away from each other that those rules shouldn't apply.
(Also, I'm just an undergrad and we really haven't covered entanglement, so this is just my rough, and quite possibly wrong, interpretation of why it's important.)
TL:DR: I guess the significance is that while the act of entangling two electrons will necessarily cause them to adopt spins opposite from one another, once you separate them out the relationship that caused them to adopt those opposite spins shouldn't apply anymore. But measurements show that they continue to obey that relationship right up until they're measured.
Edit: Made sure to specify that the rules for Fermions only apply when they're identical/indistinguishable from one another.
Heh, to be honest I can't remember a lot of that! But you're right, Bell's Theorem isn't the only expalantion for why entanglement is "spooky", since Einstein said that long before the theorem even existed. But I think it's the simplest approach for explaining to a lay audience without a lot of extra ideas or "you just gotta believe me on this bit." And I think it's what definitively killed local hidden variable theories.