Does any state give voting rights to resident aliens who have begun the citizenship process? A long time ago, many states did.
Maryland doesn’t. You have to be a citizen.
It probably doesn’t matter. I am not a citizen and have been sent a voting card. I’m sure if you registered (perjuring yourself in the process) anyone with a pulse could vote.
I didn’t vote BTW. Nor did I register but the good old State of Maryland sent me a voting card anyway. I love the USA.
Getting out of jury duty is really easy though.
The UK grants voting rights to resident aliens who are Commonwealth citizens, whether or not they have begun the citizenship process. The Soviet Union used to grant voting rights to resident aliens, at least under its first constitution. Later constitutions may have revoked that.
No state allows non-citizens to vote in state or national elections. However, some states delegate authority over school board or municipal elections to cities. Maryland is one such state, and several cities, of which the largest and best known is Takoma Park, have used this power to grant voting rights to non-citizens in local elections.
As the linked story notes, Chicago allows any parent, citizen or otherwise, to vote in school council elections.
Doesn’t the UK also allow Commonwealth citizens to be elected to Parliament without having British citizenship? 
Is the British Prime Minister actually required by law to be a British citizen?
I lived in New Zealand for several years in the 1990s and was able to vote in national and local elections purely on the basis of my residency there.
Several prefectures in Japan allow foreigners with permanent residency to vote in prefecture-level or lower elections. There are continual murmurs about expanding this nationally, but as long as the LDP is led by an Aso it ain’t gonna happen.
I don’t know. There’re rumours a snap election is about to be called, so I’ll see about registering as a candidate and let you know if I succeed. 
Yes, Commonwealth citizens can be elected to Parliament, as can Irish citizens - provided, of course, that they are UK residents. Not sure about the PM, though.
Bryan Gould , a New Zealander, was a UK MP for some time. He left politics after losing the race for leadership of the Labour Party, which if he had won, could well have seen him as Britain’s Prime Minister, all other things being equal.
In Norway, legal resident aliens may vote in local and county elections after living here for three years. Only citizens may vote in national elections.
I thought of Gould, but couldn’t find anything about whether he is a British citizen as well as a NZ citizen. Given the time he spent in the UK, he may well have been naturalised.
Or maybe not, since there is very little a British citizen can do that a resident Commonwealth citizen can’t.
The PM is just another member of parliament.
In Spain, residents from other EU countries can vote and be elected locally. This can mean up to “Autonomouos Region President” at least, not sure if they can run for National Parliament; if they can, this would mean that it’s possible to be President of Spain without being a Spaniard. Presidents aren’t elected directly, they’re selected by the corresponding Parliament.
For EU-wide elections they can vote only once.
We’ve got several town majors from other countries, no Parlamentarios yet.
Yes and no. It’s not inconceivable that special requirements could apply to that office.
I can’t find anything to suggest that’s the case, though, so I tentatively conclude that a Commonwealth citizen could indeed be PM.
Didn’t one of Australia’s PMs have delusions of become the British PM?
Many Australian PMs have had delusions on all sorts of subject matter 
I don’t think there is any formal qualification for being Prime Minister in the UK, though there are very strong conventions on the subject, including the convention that the PM be an MP.
As far as Australian PMs go, between 1901 and 1949 there was no such thing as Australian citizenship, so it would be expected that a PM be a British subject. (Generally people born in Australia during that period were British subjects, but so were people born in the UK and people born in other dominions of the British sovereign).
The only real citizenship qualification for an MP – and hence for PM – in Australia is section 44:
In Sue v Hill (1999) it was held that a dual Australian/UK citizen was disqualified under this section from being a senator. Hence a person who retains UK citizenship (or any other citizenship) along with Australian citizenship cannot be PM of Australia.
Just like the Pope – any (male) catholic could be elected as Pope. They don’t even have to be a priest; they can be ordained the same day and then raised to Bishop of Rome (Pope) – that’s happened in the past.
But doesn’t Christopher Guest (US actor) hold a seat in one of the Houses due to his inheritance of a noble title?
Yes, Christopher Guest is now the 5th Baron Haden-Guest, of Saling, Essex, acc. to Wiki. He gained the title when his father, a British diplomat, died in 1996. He only had three years in the House of Lords before the right to actually sit and vote in that body was curtailed by Act of Parliament. He has since said he’s in favor of an elected House of Lords or its equivalent.