Without disassembly, there are 43,252,003,274,489,856,000 unique permutations in a 3×3 Rubik's Cube. If sufficiently shuffled, that provides ~65 bits security.
Which means recording the colors of each of the 6 faces after two sufficient shuffles is enough to provide at least 128 bits security.
It's worth defining what it means for a cube to be "sufficiently shuffled" though. I don't know if research has been done on this (I assume so).
You need to ensure that each smaller cube has equal probability of reaching every possible location in the larger cube, as well as having each possible orientation in that position.
It looks like the shortest path to solving the worst case 3×3 cube is 20 moves. As a result, I suspect that if the cube is held behind your back and you apply 20 twists to the cube, turning it over in your hands, this would be a sufficiently shuffled cube.
@atoponce that would be my guess, though of course just because it took 20 turns to shuffle a cube doesn't mean it couldn't be solved in fewer. That may not matter in this instance, however.
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