How long has the papacy existed?

Answer

Among the Christian churches, only the Catholic Church has existed since the time of Jesus. Every other Christian church is an offshoot of the Catholic Church. The Eastern Orthodox churches broke away from unity with the pope in 1054. The Protestant churches were established during the Reformation, which began in 1517. (Most of today's Protestant churches are actually offshoots of the original Protestant offshoots.)
Only the Catholic Church existed in the tenth century, in the fifth century, and in the first century, faithfully teaching the doctrines given by Christ to the apostles, omitting nothing. The line of popes can be traced back, in unbroken succession, to Peter himself. This is unequalled by any institution in history.
Even the oldest government is new compared to the papacy, and the churches that send out door-to-door missionaries are young compared to the Catholic Church. Many of these churches began as recently as the nineteenth or twentieth centuries. Some even began during your own lifetime. None of them can claim to be the Church Jesus established.  

Another Answer

The fact of the matter is that Jesus states that: "Upon this rock I shall build my Church." This establishes Peter as the first pope. Later, Jesus establishes dogmatic law which allows the Church to change with the times and interpret scripture. Dogmatic law states that "whatever you (peter) hold true on earth will also be true in heaven." This allows popes to reform and change various teachings of the Catholic Church. It is simple fact that the Catholic Church was established by Jesus and regardless of any Protestant wishes, the Catholic Church has withstood the test of time and corruption. It is not perfect, but by participating in Catholicism a person comes most close to what Jesus gave his blessing.
I agree with you. However, I want to make it perfectly clear that Jesus pronounced a blessing on Simon, because when Jesus was asking the apostles what the people said he was, Simon stood up and proclaimed Jesus to be the "The Christ. The Son of the Living God."
Hence the blessing. "Blessed are you Simon bar Jonah, for flesh and blood hath not revealed this thing to you, but MY Father, which is in heaven. From henceforth, thou shalt be called Peter, the ROCK, and upon THIS rock, I will build my church.
And I will give thee the keys to the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven. And the very gates of hell shall not prevail against it.
The keys to the kingdom that Jesus was speaking of, and the "Rock" that Jesus referred Peter to was this.
The key was the proclamation by Peter about Jesus, and the "rock" was the steadfast belief in this. By faith, the new christian (Of Christ) believed that Jesus was the true Son of God, and that was the unmoveable guide to growing in the new faith.
And not even Satan could, would or EVER deny that.
THE Answer
How's this for "Fantasy Fable"?
THE LIST OF POPES:

1. St. Peter (32-67)
2. St. Linus (67-76)
3. St. Anacletus (Cletus) (76-88)
4. St. Clement I (88-97)
5. St. Evaristus (97-105)
6. St. Alexander I (105-115)
7. St. Sixtus I (115-125) Also called Xystus I
8. St. Telesphorus (125-136)

Another Answer/Clarification:


1. Today most non-Catholic Christian biblical scholars accept that "The Rock" is Peter himself, not Peter's testament that Jesus is the Son of God. Among biblical scholars, that fact is not in dispute. What is disputed is whether Peter's authoritative role/position in Christ's Church "passed on" to successors. And, the answer is yes, even logically yes. Jesus designated Peter as the leader/the singular authority over and above the other Apostles and Jesus did so in front of the Apostles so that there would be no mistake WHO Jesus, Himself, designated as earthly authoritative leader of His Church. To argue, then, that the Office/Chair of Peter dies with Peter is illogical, on it's face. Our Lord did build His Church on Rock, not sand or pebbles. Peter is that rock. Where Peter is, there is the Church. There have always been many voices claiming to speak for Christ. And, that is why Jesus Himself told WHO would be His voice -- Peter and Peter's successors.

2. Both Peter and Paul were executed in Rome on the same day. St. Peter was crucified upside down. Paul was beheaded. We do know, from the historical record, that Peter & Paul were both in Rome.

3. No where does the Bible (including NT) say that it is the exclusive authority. The New Testament did not even exist, as The New Testament, until 397 AD. That's practically 400 years in which the Church was preaching, evangelizing and teaching the Truth without the full, codified, New Testament scripture. There were also many scriptural writings that The Catholic Church rejected as not divinely inspired by God, i.e., The Gospel of Thomas, etc. The Church Herself does not become irrelevant once She decides which of the multiple scriptures and letter are, in fact, The Divine and Infallible Word of God. At the very least, She is the authoritative teacher of that which She produced, through the Divine Inspiration and protection of The Holy Spirit. Solo Scriptura, itself, is NOT biblical. And, St. Paul specifically tells early Christians (Corinthians I believe): "to hold fast to what WE have handed down to you by WORD and LETTER." Word is Sacred Oral Tradition. Letter is Scripture. And, this fullness of Truth is handed down by "us" (The Church speaking as ONE voice). <sigh>

4. "The Keys to the Kingdom of Heaven" that Jesus gave, exclusively, to Peter -- and thus, logically, to Peter's successors lest the church collapse, immediately, into theological confusion and chaos like we do have today in Christendom -- is very significant and scriptural. It refers back to Isaiah 22, specifically verse 15-20 but read Isaiah 22 in it's entirety. All of the Apostles (Jews) understood the meaning of The Keys and the direct reference back to Isaiah 22.
5. Pope = Papa. Papa = Father. The pope is our spiritual father -- our spiritual father in faith as are all priests. Jesus Himself described Abraham as "Our FATHER in Faith." St. Paul described himself the same as the "spiritual father" of Christians in Corinth and other areas/cities.

While the Holy Bible (OT and NT) is the infallible Word of God, the multiple interpreters of that infallible Word of God are not infallible ergo the theological chaos that exists, now, in Christendom and that has existed, most profoundly and chaotically, since the Protestant Reformation when the floodgates were opened to any and all interpretations of the Holy Bible. Each Christian denomination seems to have it's own personal, even "pet", interpretation of Scripture based on its own personal biases. Example: Luther and Calvin did not agree either and started, effectively, their own Christian churches and it's snowballed since then to include, today, multiple thousands of Christian denominations all reading the same Bible but interpreting it differently.

"ONE Lord, ONE Faith, ONE Church" -- that is the Church Our Lord established but that is clearly not what we have today. Shame on all of us.