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Why is the lowest energy so hard to find?
Frustrated Hamiltonians
Give a handful of tiny magnets some rules: these two should point the same way, those two should point opposite ways. Sometimes the rules contradict each other and one of them has to lose. Finding the least bad arrangement is easy for five magnets. For quantum magnets in general it is believed to be too hard even for a quantum computer.
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How does a quantum state fit in a computer at all?
Tensor networks
Writing down the exact state of 300 quantum spins takes more numbers than there are atoms in the visible universe. The states nature settles into when things are cold are far less tangled than that. Tensor networks write those states down using only the entanglement they actually have, which is why simulations of real materials work at all.
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Does it matter what a proof is made of?
Separations
A proof is normally something you write down and someone else checks. In quantum computing the proof can itself be a quantum state, which cannot be copied and falls apart the moment you look at it. Nobody knows whether that makes quantum proofs more powerful than ordinary ones. These are the cleanest versions of the question, and the partial answers so far.
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Is spacetime a quantum error-correcting code?
Decoding spacetime
Some physicists think space is stitched together out of entanglement, the way a hologram stores a 3D picture on a flat sheet. If that is right, getting information back out of a black hole is a decoding problem, and decoding may take far longer than the black hole lasts. Asking how long it takes needs a new kind of complexity theory, one where the input is a quantum state instead of a string of bits.
If I have got something wrong on any of these, I would like to hear about it: Kartik.anand@tuhh.de. The longer, older write-ups live in the writing.