Is there such a thing as a 'concordat' today in the Catholic church? Background to question raised:
I have been reading old histories where as a background stage, there were various antipopes (sometimes even 3 competing popes at the same time). The complex relationships they had with civil powers, kings and influence in wars and judicial inquiries and resulting punishments is like a gigantic maze.  The role of church and state becomes very confusing and messy to say the least. At times it seems impossible to figure out who the 'good guys' were amongst all the turmoil and politics among war and collecting of taxes, tithes, etc. which the kings and Pope wrestled with one another. This lead me to having an interest in the Catholic view of church and state.

Current understanding:
Upon realizing that the Catholic church believes in a physical church I have further come to understand that it sees this physical church-state to have three functions. Kingly, Priestly and Prophetic. I was aware of the priesthood and teaching role but a 'literal' kingly role, making the Pope a kind of earthly Monarch, that is having some earthly right to power (pardon me if I'm not using the correct Catholic terms) is new to me. Upon considering this 'earthly' power it started to make sense of church history. First the church had no earthly power. Then after Constantine it obviously did start having a literal earthly power. But then as Kings naturally want to separate church and state, it seems the Pope tried to make agreements with other powers so as it maintain his power in a non warring fashion under the name of a 'concordat', or a treaty between states of literal power, so to speak. However as the idea of separation of church and state, which is the way the world seems to be going since the founding of America which had multiple church denominations trying to find a new life free of religious persecution; I cant see how any state would consider the pope to have enough literal 'kingly power' as to even consider such a thing as a 'concordat'? It would seem ridiculous to even imagine such a thing, otherwise states would need to make legal agreements with every religious leader?

Resulting immediate question:
Are there any current states that still maintain some kind of official recognition of a literal state-like power of the Pope through a concordat?
Concordats in the sense of a "treaty" were prominent in Pope Pius XI's papacy (cf. the Lateran Treaty) and in Pope Pius XII's also (Pope Pius XII was Pope Pius XI's secretary of state). See this list of concordats. 

Also, there are some good books on the Catholic Church and State doctrine:


Catholic Church and Christian State: A Series of Essays on the Relation of the Church to the Civil Power vol. 1 &amp; vol. 2 by a cardinal-archivist of the Vatican Archives, Cdl. Joseph Hergenröther
The Temporal Power of the Vicar of Christ by Cdl. Manning
Catholic Principles of Politics by Fr. John A Ryan