Apparently the degree of entanglement between distinct vacuum states falls off as the distance between them. But perhaps this can be inverted, so that the concept of distance could be seen as an emergent proxy for the degree of entanglement.
The 50 minute lecture is 'aimed at undergraduates who haven't necessarily yet taken a quantum mechanics course'. If you are such, Carroll's talk will be as compelling as a presentation on Summa Theologica from Thomas Aquinas.
On the other hand, a passable familiarity with Hilbert space, quantum superposition and the Schrödinger equation plus a hand-wavy feel for QFT and Einstein's field equations will allow you to properly appreciate Carroll's approach to physics (and would make you a physics graduate).
In a nutshell, it's believe in the maths. Once you appreciate the ubiquity of superposition (ie, it's everywhere) you're kind of committed to the reality - in some sense - of Hilbert space. The observed phenomena simply can't be explained by theories which restrict themselves to our classical-looking 4D spacetime.
Carroll's talk is not technical in argumentation, he mentions rather than uses the theoretical apparatus of modern physics. That does put the burden of getting his drift wholly on the theoretical preparation of the listener of course.
In the final part of his lecture, he describes the research programme which seeks to obtain geometry from entanglement in quantum field theories via entropy and then, through considerations of energy, to reconstruct the GR field equations as the classical limit.
He seems encouraged, though this is work-in-progress.
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