What is the successor state to Ancient Rome/SPQR

The OP said Rome=SPQR.

Not the Roman Empire, but the country of the Romans.

Therefore its wrong to consider “rights and obligations”. Thats what defines an empire, a country is defined differently.

The Roman country was a combination of the two alliances, The first the Roman cities, and the 2nd the italian cities, which pretty much dominated and thus defined a country , the Roman Republic run by SPRQ.

The alliances of cities lasted until the 1500’s, and then came the times of the Lombard, Spain, HRE hegemony … cities lost the power to their surrounding states, but the basic power block remained in southern Italy which regained the north to form Modern Italy.
In many ways, including even the boundaries between France, HRE/Deutchland, the loss of identity to Spain or HRE is not consequential enough, the system of alliances of cities formed a country that maintained the population and the culture, and that can be seen in modern Italy

What about one non-state polity that unifies 3 of the Western contenders and a lot of Imperial territory - the European Union.

IIRC William considered Harold to be a usurper, and not a legitimate King. Therefore WC would have viewed himself as the successor to Edward the Confessor, after whose death Harold had illegally claimed the crown.

It didn’t exist until a half-century ago (by the most generous interpretation), so some other entity had to succeed to the Roman Empire first. You can’t inherit something that didn’t exist for a while.

I’d imagine, notwithstanding the possibility of being killed for being an idolater etc. a regular Roman citizen would find himself more at home under the Abassids, than any of these other suggestions, barring perhaps the early Byzantine periods, whilst it at least still oriented itself west, perhaps under Justinian.

The idea of a successor state is a tricky one.

The European Union.
If you ask a Eurocrat.
Nobody else thinks that.

The EU is clearly the Fourth Reich.

since this is the gq and it is clear you are not a person who is spending times in the halls of bruxelles, perhaps refrain from not informed political comment?

By my remark, I meant that the EU has unified Western Europe, as Rome has.

Allegedly.
I have a few doubts.

ASt no time do I view the EU as an “empire”.

Didn’t Nostradamus write a quatrain about this?
Something about a blue turban and the EU flag is blue?

Maybe.
Ottoman Empire/Turkey[edit]
There is some debate over whether the modern Republic of Turkey was the continuing state to the Ottoman Empire or a successor.[3] The two entities fought on opposing sides in the Turkish War of Independence (1919–1922), and even briefly co-existed as separate administrative units (whilst at war with one another): Turkey with its capital in Ankara and the Ottoman Empire from Istanbul. The nationalist faction, led by Mustafa Kemal who defected from the Ottoman army, established the modern republic as a nation state by defeating the opposing elements in the independence war.