The diamond must become quantum as a unit, and the wave function of the quantum diamond must then disperse sufficiently to extend outside the box. At that point the diamond as a whole unit has a probability of jumping outside the box.
The first criterion is by far the most difficult, because it can only be achieved by keeping the diamond in total and absolute information isolation from the rest of the universe. That is... unlikely to say the least. If even a single photon or phonon "detects" its location, then from that point forward the diamond is classical in the sense that the photon has pinpointed its location.
The second criterion is just abysmally slow. Because even a small diamond has a lot of mass, its wave function disperses very, very slowly.
Both of these criteria can be expressed in terms of path integrals, which provide a precise way to quantify the issues I only described conceptually.
@OllyPrice very reasonably asked for clarifications on:
(1) what does total isolation from the universe mean?, and
(2) what does it mean that the diamond must be kept quantum as a unit?
The most concrete way I can think of to quantify isolation is that the diamond can neither emit or receive particles of matter or energy.
Keeping out particles of matter is the easier part of that, since it means you "just" need the most absolute vacuum every created, including removal of all high-energy particles such as cosmic rays. Preventing energy exchanges is much, much harder. The vacuum keeps you from exchanging phonons (sound quanta), so you get two (matter and phonons) for the price of one with that one. Your suspension system would need to be phonon-free, however, at least if you do the experiment here on earth.
That leaves mostly electromagnetic radiation. Radio frequencies whose wavelengths are a lot larger than the distance you want the diamond to jump are not a big problem, although if you get enough of them you may start locating the diamond too well and thus "lose coherence" as they say these days (it's the same idea).
So, that leaves mostly the higher frequencies of very short microwaves through infrared and light. Infrared is going to be the biggest issue, so you want both the diamond and the cavity in which you do the experiment to be cold, as close to absolute zero as possible, to prevent stray space-locating infrared photons from traveling in either direction.
... and having said all that, I must also say: Hmm! That portfolio of preconditions is not quite as impossible as I had always assumed. So, someone could possibly do this for real in a high-end lab, using something like, say, a bacterium (huge!)... or much better, something a lot smaller, such as a low-end nanodiamond. The suspension system for the earthbound version would be the trickiest part. Hmm. Maybe a single embedded electron charge? No, much easier: A superconducting particle over a dish magnet. Or maybe even better: A very small piece of pyrolytic carbon similarly suspended by a magnetic field, though I don't know for sure what would happen to that materials extreme form of diamagnetism at such low temperatures.
That is a truly fascinating possibility! Does anyone here know if anyone has ever tried this?
I certainly don't know. But wow, what a fascinating possibility: A material object, albeit a very very very tiny one, quantum-jumping through a physical barrier? Now that would be a thesis paper!
On to question (2), what makes the diamond a "unit"?
That one's easy! Bonding. The covalent bonds of the diamond keep it internal components aligned strongly with each other, and so unable to deviate over time into less certain relationships. That's not to say you can't get some weird stuff going on internally, but it won't in general be capable of disrupting the diamond's internal structure.
Notice the inverse relationship here between isolation and cohesion (bonding).
That is, anything that bonds two objects together physically (phonon-mediated) or via information keeps them from "going quantum" relative to each other. Conversely, it is the lack of such bonding (isolation) that enables relative quantum behavior.
When the universe as a whole is one of those two partners, "relative quantum behavior" becomes a pretty one-sided concept, since we in the universe just stay classical.
However, for something like two very small objects isolated both from each other and from the universe, the concept of relative quantum behavior becomes a testable idea. That would be where each system sees the other as being the quantum one, whenever they finally do interact with each other.
(Fair warning: I just now made up the phrase "relative quantum behavior"; it's not standard. However, the fully quantum frameworks for studying systems such as positronium would necessarily contain an equivalent concept, since for example in positronium the electron and positron are necessarily equally "quantum" relative to each other, whereas in hydrogen it's easy to approximate the proton as being classical. But I don't know if the idea has ever been explored as a stand-alone concept, especially as it would apply to larger systems.)
Finally, and even more interesting (to me at least) is the idea that might be able to encapsulate a "hot spot" within a sufficiently large, cold piece of matter. By recording over time, this internal observer could watch itself even as the diamond as a whole "goes quantum" and starts doing things like being in two places at once.
The idea that a classical observer could nonetheless still be subject to quantum non-locality at a larger level just fascinates me, in part because it's so counter-intuitive to how we usually view quantum mechanics.