The term "quantum jump has entered everyday language as a metaphor for a large, discontinuous change. It has also become widespread in the vast but sadly repetitive landscape of pseudo-science and mysticism.
The term comes from physics, and is indeed used by physicists (though rarely in published papers). It evokes the fact that mutually distinguishable states in quantum physical systems are always discrete. Yet there is no such phenomenon in quantum physics as a "quantum jump": under the laws of quantum theory, change is always continuous in both space and time. OK, maybe some physicists still subscribe to an exception to that, namely the so-called "collapse of the wave function" when an object is observed by a conscious observer. But that nonsense is not the nonsense I am referring to here. I'm referring to misconceptions even about the sub-microscopic world—like: "when an electron in a higher-energy state undergoes a transition to a lower energy level, emitting a photon, it quantum-jumps from one discrete orbit to another without passing through intermediate states".
Even worse: "when an electron in a tunnel diode approaches the barrier that it does not have enough energy to penetrate (so that under classical physics it would bounce off), the quantum phenomenon of tunneling allows it to appear mysteriously on the other side without ever having been in the region where it would have negative kinetic energy".
The truth is that the electron in such situations does not have a single energy, or position, but a range of energies and positions, and the allowed range itself can change with time. If the whole range of energies of a tunneling particle were below that required to surmount the barrier, it would indeed bounce off. And if an electron in an atom really were at a discrete energy level, and nothing intervened to change that, then it would never make a transition to any other energy.
Quantum jumps are an instance of what used to be called "action at a distance": something at one location having an effect, not mediated by anything physical, at another location. Newton called this "so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it". And the error has analogues in fields quite distant from classical and quantum physics. For example in political philosophy the "quantum jump" is called revolution, and the absurd error is that progress can be made by violently sweeping away existing political institutions and starting from scratch. In the philosophy of science it is Thomas Kuhn's idea that science proceeds via revolutions—i.e. victories of one faction over another, both of which are unable to alter their respective "paradigms" rationally. In biology the "quantum jump" is called saltation: the appearance of a new adaptation from one generation to the next, and the absurd error is called saltationism.
Newton was wrong that there is a maximum size of error that competent people can fall into, but right that this particular one is severe. All those versions of it are mistaken for a single reason: they all require information of the requisite kind to appear from nowhere. In reality, the space on the far side of the barrier cannot "know" that an electron, and not a proton or a bison, must appear there, until some physical change, originating at the electron, reaches it. The same holds when it is not a spatial gap but a directly informational one: Political institutions, and biological adaptations, instantiate information—knowledge—about how a complex system can better meet the challenges facing it, and knowledge can be created only by processes of piecemeal variation and selection. And Kuhn's vision cannot explain how science has in fact been delivering knowledge about physical reality at an ever-accelerating rate.
Quantum jumps in all these fields represent a retreat from explanation, and therefore in effect an appeal to the supernatural. They all have the logic of the Sidney Harris cartoon "Then a Miracle Occurs" (depicting a mathematician with a gap in his proof). As Richard Dawkins puts it, "saltationism is creationism". And in all cases the reality that fills the gap, the idea that truly explains the phenomenon, is much more interesting and delightful than any faith in its mystery could be.