Etymology of "quantum leap"

As suggested by the following source, it refers to the connotation of quantum meaning a "sudden large increase".

Quantum leap:

• This is a transferred or figurative use of the concept of the same name in particle physics or quantum mechanics. From the OED: "quantum increase, a sudden large increase; cf. quantum jump; quantum jump, an abrupt transition between one stationary state of a quantized system and another, with the absorption or emission of a quantum; also transf., a sudden large increase or advance; quantum leap, a sudden large advance.

(The Phrase Finder)

The following source illustrates the evolution of "quantum" meaning "something big":

• The first use of “quantum leap” to mean “really big” was in 1956, the OED says, in a discussion of the US-Soviet balance of power in a nuclear postwar world, where a writer described “The enormous multiplication of power, the ‘quantum leap’ to a new order of magnitude of destruction.”

But note that:

• Though using “quantum leap” to mean “big jump” is fully idiomatic, it’s best to avoid using just plain “quantum” to mean “huge” (especially if addressing a physicist). Most dictionaries still define it as simply “an amount.” It’s only the hyperbole that gives it mass quantity.

(archives.cjr.org)

Quantum does not mean "small and discrete", it only means "discrete", as in "quantisation" - fitting values into a discrete defined set. For instance in mathematics the operation of "rounding" quantises fractions into the set of integers.

Similarly in electronics, sampling signals quantises them into a discrete digital value.

You have the connotation of "small" because quantum physics coincidentally mostly deals with very small things - the smallest divisions (or quanta) of matter.

A quantum leap, however, has the connotation of large, because a leap covers a large distance. However it is quantum, so cannot be further divided within the context that it is used.

For instance the United States could have all its coordinates quantised to within the individual states. A step from the center of one state to the center of a neighbouring state is quite a large distance, but it is only a change of one quanta.

So a quantum leap is a discrete change from one state (not a US state now) to the next state, with the connotation that the states involved are big and cover a large "area".

Another good way of thinking about it is a staircase. Each step is a quanta. The bigger the steps the bigger the leap to get to the next one.

A good synonym is also "step change".

In the standard model of an atom a Quanta means discrete steps or values. An electron can only be in any one discrete size of orbit. If it is made to move out of this orbit it will leap to the next discrete orbit.