According to R' Moshe Feinstein as cited in this answer, cheating on a school test/examination can be considered theft from a future employer. E.g. if you get the degree by cheating on a math test, you are tricking a future employer into thinking that you know math when you really don't.
If a Jew is doing Teshuva for a prior instance of cheating on a test and wants to make sure that they avoid stealing from any employer in the future, what must they do to fix their situation? Is it sufficient for them to learn the missing material? Would they need to do more?
E.g. suppose Reuven cheated on his high school exit exam on the section covering quadratic equations by using crib notes to cover up his abysmal algebraic competencies. He now wants to stop stealing from employers by tricking them into thinking that his high school diploma demonstrates that he knows quadratic equations. Which of the following (if any) would he need to do?
1) Reuven just needs to learn quadratic equations.
2) Reuven needs to learn quadratic equations and get his quadratic equation knowledge tested by a trustworthy external source (e.g. he could hire an independent teacher to test him).
3) Reuven must pass the exam that he cheated on (or at least a substantially equivalent exam), but it can be graded by any reliable teacher, not necessarily the same teacher who originally gave the exam, one with authority to change Reuven's original grade or transcript, or even one affiliated with the educational system in which the cheating took place (e.g. if the exam was from New York, the exam could be re-taken in France and graded by a French teacher who happened to be competent in the subject matter of the original test but had no affiliation with any New York educational system or authority to act for them).
4) Reuven must go back and formally re-sit the exam, possibly re-enrolling in high school math coursework if the coursework must be re-taken in order to formally re-sit the exam.
5) Reuven must re-enroll in high school (or enroll in a high school equivalency program) and earn a new diploma, but he can apply for/accept prior learning credit for coursework and exams on which he did not cheat.
6) Reuven must restart high school from the very first day and/or complete an equivalency program from start to finish with no credit for prior coursework.
In response to DanF's comment, an answer could address whether the gravamen of the halachic issue is in falsely claiming mastery of academic material over which one does not actually have mastery (and which the solution would be to simply learn the material), or whether the issue is really a procedural violation with respect to the school that can only truly be rectified by going back to the school or instructor, confessing, and formally re-sitting the applicable exams/coursework?
Obvious practical problems have been mentioned in the comments. This is not only a problem with the Regents' exams - the vast majority of school assessments cannot easily be "retaken" later a la carte. It seems that many English A-Levels can actually be re-sat as an adult, but that seems to be an exception to the general rule. If I called up the Baltimore public school system and said, "Hi, I think I cheated on Unit Test #2 for Algebra 2 in 1985, could you guys have a copy ready when I arrive tomorrow so I can retake it?" they would simply laugh at me and tell me to move on with my life, or possibly tell me to pick up a Teach Yourself Algebra book or register for a community college course if I was feeling really bad.
This is also not solely a Jewish issue - see this related question on Academia.SE, in which one of the answers mentions
...trying to attain 100% of anything is an example of what is known as a category error -- the act of attributing a concept that is applicable to one realm of thought to another realm where it is no longer applicable. "100% ethical standing" is simply a meaningless concept.
which raises the question of whether this falls into one of those "how the market works" exceptions - that since employers already know (or should know) that most job applicants have previously engaged in some minor amount of cheating in school, then there is no גניבת דעת unless one's cheating was significantly greater than most.