A Kerr-AdS black hole is eternal, never evaporates and has a Malament-Hogarth metric. Bob, a universally programmable reversible classical computer with a fixed maximum memory who only outputs one bit, orbits around the black hole for a superpolynomial time. Bob aspires to be a universal computer. He accepts any program for a universal computer and tries to compute it, but doesn't always succeed because he might not survive long enough, but is never wrong if he manages to output an answer. Alice selects an $l$ bit program of her own choosing, programs Bob with it and initiates the program. She commits a copy of her program to her memory permanently and will never forget it. Then, she falls through the outer event horizon. In the Penrose diagram of the Kerr-AdS black hole, there are actually two different inner event horizons Alice may fall through afterward. There's also another entangled AdS double universe to prevent firewalls from forming. Only one of the two inner horizons has the Malament-Hogarth property, and Alice chooses that one to fall through. Alice has a rocket so she doesn't have to follow a geodesic.
Meanwhile, Bob computes the superpolynomially long reversible program Alice chose. Reversibility means Bob doesn't have to dump a superpolynomial amount of entropy into his environment according to Landauer's principle. Actually, the volume Bob can take up goes exponentially in $kr$ where $r$ is the radius of Bob's orbit and the number of computational steps per unit coordinate time also grows exponentially. Geodesic orbits no longer exist if $kr$ is large enough, but we can let Bob be a spherical shell of radius $r$ totally surrounding the black hole instead. As a spherical shell instead of a satellite, Bob doesn't have to worry about spiraling inward after radiating gravitational waves or Brownian motion from the Hawking heat bath. The mass density of Bob then needs to be low enough so that Bob doesn't collapse into a black hole of his own. However, the energy contribution of Bob to the universe can be much greater than the mass of the black hole. Despite being a classical reversible computer, Bob can still broadcast internal information because classical information can be cloned.
At the end of the computation, Bob deputizes Carol, who also has a rocket, to catch up with Alice and inform her about the output bit. So, Alice learns about the output of the program of her choice after experiencing a linear amount of subjective time. Alice doesn't come to an end at a singularity, so black hole complimentarity isn't needed, but why would black hole complimentarity suddenly apply in the limit of the angular velocity going to zero? Can the CFT dual of this future AdS universe perform some superpolynomial computations?
Well, Bob can't compute indefinitely due to being immersed in a Hawking heat bath to equilibrate with, but for how long can he remain computing? If Bob's temperature is much larger than the Hawking temperature, what's the heat conductivity between Bob and the black hole? Or is Bob's temperature the Hawking temperature?
Even without Bob, Alice can make some PP computations with a different program of her choice by making use of closed timelike curves after passing through the Cauchy horizon of the timelike ring singularity. Can this show up in the CFT dual of this future AdS universe?
Actually, blue shifting of generic small perturbations are likely to prevent inner horizons from forming in the first place due to gravitational back-reaction causing a spacelike singularity to develop instead, but is this necessarily true for all possible quantum states of the entangled universe and its double? After all, that's not what we expect if we have locality at inner horizons. Just as we can have a thermodynamic past hypothesis at the big bang, we might also have a thermodynamic future hypothesis leading to a local reversal of the thermodynamic arrow of time. (Speaking of which, if AdS has no past hypothesis, then why can it have a thermodynamic arrow of time?) The Hawking temperature of the inner horizon differs from that of the outer horizon. If there are exceptions, can one such quantum state be described by a polynomial amount of classical information? After all, if such a state exists, I just described it in a few words together with Alice' $l$ bit program!
Also, there's a lack of global hyperbolicity right after Alice and Carol pass the inner horizon.
Even if a singularity definitely prevents inner horizons from forming, Carol might still use her rocket to catch up with Alice an inverse superpolynomially small time before Alice hits the singularity. This is much smaller than the Planck time, so Planck scale dynamics might modify this description. Black hole complementarity now has to apply unless superluminal signaling is admitted. So, can the stretched outer horizon compute some superpolynomial computations? What about the combined CFT dual of our universe and its double?
Actually, Alice, Carol and Bob can't be classical because there's no invariant decoherent pointer basis for any of them. So, let Alice carry $l$ qubits, Carol carry a qubit and Bob be a quantum computer instead. Prepare Alice' $l$ qubit program to be entangled with Bob, and Bob hands over the output qubit to Carol. Actually, quasiclassicality remains possible if Alice, Carol and Bob store a huge number of redundant copies of their information, as an error correction code. If Bob’s program were quantum, there can only be one copy of it throughout all of Bob. If it were classical, there can be numerous copies of it all over Bob.
As Landauer once said, information is physical.