I would like the idea of voting diplomas, but ensuring that such gate-keeping can evolve over time without being gamed to slowly strip a larger & larger fraction of the population of representation over time is a really tricky problem.
Thus, I think that the freedom to vote & to hold office should be enshrined in the constitution for everyone, regardless of criminal history or citizenship.
It would take a similarly isolated gov department to rate and set the difficulty. That one should probably not have a diploma.
That the system will be gamed is no real concern as that battle will continue forever. This version is at least capable of plugging the hole when enough people notice it.
The freedom to hold the generic office seems fine. For a specialized department some relevant experience and diplomas isn't a lot to ask. If unskilled employees could do a great job in any occupation we would see this in the private sector. I think it can just be left for judgement by the voter.
Thus, I think that the freedom to vote & to hold office should be enshrined in the constitution for everyone, regardless of criminal history or citizenship.