My first two summaries were:
b. Spinning Space Buckets
c. Relativity and the Absolute
1. This chapter seemed a lot longer than it needed to be to me. Hopefully I can give the gist fairly quickly. When you run a water wave through two openings, you will get an "interference pattern" on the other side of the openings. The same happens with light. When you shine laser light through two slits, you get the same interference pattern.
What is very, very strange is that if you take an electron beam and shoot electrons one by one, slowly at those same two slits. If you shoot them, each one separately so they do not interact with each other, over time the very same interference pattern will emerge. So electrons, photons, all matter may be made up of particles, but those particles behave like waves.
The kind of wave it is, Greene helpfully points out, is a probabilistic wave. That is to say, it is because particles have a greater or lesser probability of being at a particular place when interacted with, over time their interaction with the slits plays out as a distribution of lines that fits those probabilities.
2. Werner Heisenberg showed in the late 20s that you cannot measure both the position and velocity of a particle accurately at the same time. If you measure the position with precision you can't measure the velocity and vice versa. This also applies to a number of other atomic features, such as spin.
Einstein engaged in a longstanding debate with the quantum mafia led by Bohr. Einstein couldn't bring himself to believe what has more or less turned out to be true. Particles don't actually have a definite position or velocity until you measure them. The nature of the quantum world is probabilistic. There is a greater or lesser probability that an electron is somewhere. It's not that it is somewhere and we just don't know exactly where. It's that it isn't exactly somewhere.
3. Einstein and a couple colleagues unintentionally advanced this discussion with a thought experiment that was later carried out. He suggested that if two twin particles parted with a correlated identity, as is often the case, then by measuring the position or the velocity of the one you could indirectly infer the position or velocity of the other.
This seems like common sense. What you do to the one doesn't affect what you do to the other, so you can measure the one and not disturb the other. David Bohm extended the Einstein thought experiment to the spin of a particle. In theory, if you measure the spin of a pair of correlated particles, you should be implicitly identifying the spin of the other.  In other words, the other one would have a definite position and velocity even without measuring it, contrary to what the Copenhagen mafia insisted.
4. In the 1960s, Jon Bell came up with a way to see if Einstein was correct and in the 70s and 80s, it became possible to test it. He determined that if you randomly measured the spin of two correlated particles in relation to more than two possible states, you could determine whether both particles had a definite spin to begin with. If you randomly measured the spin at three different angles for both particles, those measurements would agree more than 50% of the time if both of them had a definite spin to begin with.
Some of the best tests took place in the early 80s by the French scientist Alain Aspect. He showed that the detectors did not show that the spins agreed more than 50% of the time. What they showed was rather astounding.
- If Einstein had been correct, they would have agreed more than 50% of the time. The implication would be that the particles had a definite state before measurement, as Einstein thought must surely be the case.
- If the quantum mafia had been completely right, the measurements would have agreed less than 50% of the time.  The implication would be that the particles had an indefinite state before measurement and randomly took a state when measured.
- What happened is that they agreed exactly 50% of the time. The implication was that they had an indefinite state before measurement but both particles took on the same state when one of them was measured.
5. This is called quantum entanglement. What you do to a particle here can affect a particle there, no matter where "there" is. In a sense, there is no such thing as "locality" in space. There is no "here" that is distinct from "there."
It's not that one particle sends a message somehow to the other, correlated particle. They rather have a unity that transcends locality. Special relativity is not violated. Nothing moves faster than the speed of light. It's just that there is a synchrony that transcends space.
 BTW, Bohm fled the US in the middle of the McCarthy nonsense and ended up in England at the end of his life. I hope America will never have a witch hunt like that again. Just think of how many brilliant minds Hitler lost in the middle of his ideological nonsense. No country can afford to lose its scientists for whatever stupid reason the public or politicians come up with.
 The Copenhagen circle with people like Niels Bohr, Werner Heisenberg, and Wolfgang Pauli were positivists in philosophy. They didn't consider anything to be real if you couldn't measure it. Their way of explaining the uncertainty principle is deeply unsatisfying to me. Although Einstein proved to be wrong, his objection to them was perfectly valid. Just because you can't measure something doesn't necessarily prove it doesn't exist.