Ignoring the ethics: what reason would you have to answer truthfully? Someone could easily circulate the "correct" responses, and then you pass the barrier with basically no difference. What are you going to do, eliminate voting anonymity and chase people up afterwards to see if they can actually answer the way they said they did?

One form of question could be this. Then even if people parrot the right answer, they are still being improved and have to acknowledge their profound wrongness every time they visit the ballot box.

Your child is very sick. Do you

(A) Do what is recommended by peer-reviewed scientific literature, of which one of the central tenets is that you only believe things which are backed up by data and solid arguments.

(B) Do what is recommended by your religion, of which one of the central tenets is that faith is sufficient reason to believe in something, and neither data not solid arguments are needed.

So which is the "right" answer, in your system?

And who gets to determine (and enforce) "rightness"?

The system that determines "rightness" is the system that makes your medicines work, your planes fly reliably, your bridges safe to drive over, and the computer and communications network work that allowed you to type that silly question into.

So you would categorically exclude anyone who disagrees with you?

Anyone who thinks that religious faith is a better route to knowledge than science, yes. You wouldn't?

someone will quickly sneak in stuff like "is global warming real?" in the questions.

at least "What is the European Union?" when voting for brexit would be enough.

That's a good (and presumably unintentional) demonstration of the problem - the European Union is a hugely complex, very political beast and the right answer to that question depends heavily on your political frame of reference and priorities. Even just the free trade-related aspects can be accurately described either as giving up control of our own laws and regulations to a centralized bureaucracy or enabling trade by removing non-tariff barriers depending on how you want to frame them - since almost all non-tariff barriers other than differences in regulations are banned by WTO rules they're the same thing, one description just emphasises the benefits and the other the downsides. (I've even seen a certain FT writer who insists he's not pro-EU use both framings - the positive one when discussing the EU and the negative one when discussing post-Brexit trade deals outside the EU.)

True, but maybe there could be some sort of democratic consensus system to establish the questions (one which by design avoids the superficial circularity).

Ask questions that a 5th grader could answer. The level does not hove too be that high for it to be effective.

Or... just ask science questions: "Is planet Earth flat?".

That just tests the ability to stick to consensus - what happens when you get the equivalent of "does the Sun orbit the Earth" in a society before Copernicus and Galileo? You also move away entirely from the original point of the questions - someone with scientific knowledge doesn't necessarily know anything about society and politics. They're separate worlds.

In a society before Copernicus and Galileo that question would have been cutting edge science. So just stick to well understood things.

Even that is apparently fraught because a few people (though no sects I'm aware of) profess belief in a flat Earth for religious reasons. For example, Biblical literalists interpreting Job 38 as implying a flat Earth ("Where wast thou when I laid the foundations of the earth? [...] Whereupon are the foundations thereof fastened? or who laid the corner stone thereof").

That's exactly the point: exclude those people. The idea is not to have everybody pass the test.

You have been downvoted, but I too would think that was a good idea - unfortunately there are many potential traps here, not the least of which is the ability to control the questions.

In addition to that, you end up breaking the democratic system, which is so very good at keeping our disputes and disagreements from boiling over into a civil war.