Sometimes, there is an absolute limit on the number of terms an officeholder can serve, while, in other cases, the restrictions are merely on the number of consecutive terms.
For example, Maxwell's Equations would always be capitalized, while quantum mechanics is usually un-capitalized. I searched half a dozen sites for an answer to this but found no clue about a rule. So go either way that makes you happy, but (most of all) be consistent.
The word quantum refers to the fact that some quantities do not come in a continuous range of values, but instead have multiple possible discrete values. The integer quantum numbers you refer to are just an arbitrary assignment of an integer to a given discrete value.