when we measure some measurable value the wave function collapses immediately everywhere.

When I studied physics (30 years ago), this concept stopped me from believing what was taught; because it is clearly wrong. Professors couldn't even answer the question "then what IS a measurement"? At the time it was even thought to have to do with humans observing. At the same time, a universe wide instantaneous change because a human observes something was as indigestible for me as it was for Einstein.

Ever since I have given this deep thought and the only satisfactory answer that I could come up with is the following (which turns out to already exist as the Many Worlds interpretation):

Consider the simplest of cases where two particles have just two possible eigen states (classical states, or measurable states): spin up or spin down. This as opposed to a continuous wave function, but the idea is the same. Furthermore, assume these particles are entangled so that the only possible states of the combined system of the two is {up,down} and {down,up}. Then, before measurement there are two possible future realities: either you measure up for the first and down for the second, or you measure down for the first and up for the second. You don't know which of the two cases that will be, but you know (knowing the super position state and entanglement since this is how the system of two particles were created) that you will never measure {up,up} or {down,down}.

My interpretation was then that these two (future) realities are indeed two realities: two real "time-lines", which I called "realities". A reality however is purely linked to an observer: the reality is what you perceive as such. Other people perceive different realities (even just in the context of general relativity), quantum "realities" however are orthogonal: they do not interact.

So, the state of the system is a superposition, but you can think of it as a vector that can be projected onto two axis to get two coordinates, where the axis are the (two) possible realities.

This solves two things: 1) there is no collapse of wave functions necessary anymore at all. 2) there is no need for anything to go faster than light.

Consider sending one of the particles away to a friend who is a light year away. And then "at the same time" you measure each your particle. In both cases you become entangled with the particle you measure (a process bound by the speed of light) and your reality seems to split into two: there is now you who measured up and a you who measured down. This is simply propagation of the Schrodinger equation, without talking about "measurement" or "wave function collapse". It is much more simple and intuitively correct. However, these two realities are orthogonal and can not interact. You yourself will on be conscious of one of them, and to you it seems you either measured up or down, with a 50% chance to find yourself in either of the two realities.

Your friend does the same and also finds himself in one of two possible realities. The fun part is: there were only two realities to begin with (with respect to the eigen states of these particles that is): {up,down} and {down,up}. One of the realities that your friend is in CAN interact with one of your realities, and the other two realities too. Your friend when traveling slower than light to you in either or both of his realities will meet you in a reality where he measure up and you down, or in a (now shared) reality where he measured down and you up.

The full realization of how this works came to me only 25 years later, when I wrote a program to simulate a quantum computer: in this program I had to write out the exact formulas and mathematics; and what turned out? I could reuse the code for entanglement for the measurements: they are EXACTLY the same thing!

Measurement == entanglement.

There is no wavefunction collapse and nothing going faster than light. We'll just have to accept that what we perceive as "reality" is just that what propagates in a consistent way over larger distances and time (ie, macro scopic) and therefore is limited to eigen values, but that there is in fact a universe-wide quantum wave that is evolving according to the Schrodinger equation that describes many many such realities at once.

PS My quantum computer simulation can be found here: https://github.com/CarloWood/quantum and a measurement node is the same (just slightly changed to see the difference in the output) as a controlled-NOT gate: https://github.com/CarloWood/quantum/blob/master/src/Circuit.h#L122