No other Church can legitimately make this claim.
These early writings confirm the truth that Jesus started one Church and it was the Catholic Church.
First up, this is taken from the letter of Pope Clement, the fourth pope, written to the Corinthians in about 90 AD. I've highlighted some key quotes that demonstrate the authority of the Church and the pope.
First Epistle to the Church at Corinth,
by His Holiness Pope St. Clement I
Chapter 40
These things therefore being manifest to us, and since we look into the depths of the divine knowledge, it behoves us to do all things in their proper order, which the Lord has commanded us to perform at stated times. He has enjoined offerings to be presented and service to be performed to Him, and that not thoughtlessly or irregularly, but at the appointed times and hours. Where and by whom He desires these things to be done, He Himself has fixed by His own supreme will, in order that all things being piously done according to His good pleasure, may be acceptable to Him. Those, therefore, who present their offerings at the appointed times, are accepted and blessed; for inasmuch as they follow the laws of the Lord, they sin not. For his own peculiar services are assigned to the high priest, and their own proper place is prescribed to the priests, and their own special ministrations devolve on the Levites. The layman is bound by the laws that pertain to laymen.
Chapter 41
Let every one of you, brethren, give thanks to God in his own order, living in all good conscience, with becoming gravity, and not going beyond the rule of the ministry prescribed to him. Not in every place, brethren, are the daily sacrifices offered, or the peace-offerings, or the sin-offerings and the trespass-offerings, but in Jerusalem only. And even there they are not offered in any place, but only at the altar before the temple, that which is offered being first carefully examined by the high priest and the ministers already mentioned. Those, therefore, who do anything beyond that which is agreeable to His will, are punished with death. You see, brethren, that the greater the knowledge that has been vouchsafed to us, the greater also is the danger to which we are exposed.
Chapter 42
The apostles have preached the Gospel to us from the Lord Jesus Christ; Jesus Christ has done so from God. Christ therefore was sent forth by God, and the apostles by Christ. Both these appointments, then, were made in an orderly way, according to the will of God. Having therefore received their orders, and being fully assured by the resurrection of our Lord Jesus Christ, and established in the word of God, with full assurance of the Holy Ghost, they went forth proclaiming that the kingdom of God was at hand. And thus preaching through countries and cities,
they appointed the first-fruits of their labours, having first proved them by the Spirit, to be bishops and deacons of those who should afterwards believe. Nor was this any new thing, since indeed many ages before it was written concerning bishops and deacons. For thus says the Scripture a certain place, "I will appoint their bishops in righteousness, and their deacons in faith."
Chapter 43
And what wonder is it if those in Christ who were entrusted with such a duty by God, appointed those ministers before mentioned, when the blessed Moses also, "a faithful servant in all his house," noted down in the sacred books all the injunctions which were given him, and when the other prophets also followed him, bearing witness with one consent to the ordinances which he had appointed? For, when rivalry arose concerning the priesthood, and the tribes were contending among themselves as to which of them should be adorned with that glorious title, he commanded the twelve princes of the tribes to bring him their rods, each one being inscribed with the name of the tribe. And he took them and bound them together, and sealed them with the rings of the princes of the tribes, and laid them up in the tabernacle of witness on the table of God. And having shut the doors of the tabernacle, he sealed the keys, as he had done the rods, and said to them, Men and brethren, the tribe whose rod shall blossom has God chosen to fulfil the office of the priesthood, and to minister to Him. And when the morning was come, he assembled all Israel, six hundred thousand men, and showed the seals to the princes of the tribes, and opened the tabernacle of witness, and brought forth the rods. And the rod of Aaron was found not only to have blossomed, but to bear fruit upon it. What think you, beloved? Did not Moses know beforehand that this would happen? Undoubtedly he knew; but he acted thus, that there might be no sedition in Israel, and that the name of the true and only God might be glorified; to whom be glory for ever and ever. Amen.
Chapter 44
Our apostles also knew, through our Lord Jesus Christ, and there would be strife on account of the office of the episcopate.
For this reason, therefore, inasmuch as they had obtained a perfect fore-knowledge of this, they appointed those presbyters already mentioned, and afterwards gave instructions, that when these should fall asleep, other approved men should succeed them in their ministry. We are of opinion, therefore, that those appointed by them, or afterwards by other eminent men, with the consent of the whole Church, and who have blame-lessly served the flock of Christ in a humble, peaceable, and disinterested spirit, and have for a long time possessed the good opinion of all, cannot be justly dismissed from the ministry. For our sin will not be small, if we eject from the episcopate those who have blamelessly and holily fulfilled its duties. Blessed are those presbyters who, having finished their course before now, have obtained a fruitful and perfect departure [from this world]; for they have no fear lest any one deprive them of the place now appointed them. But we see that you have removed some men of excellent behaviour from the ministry, which they fulfilled blamelessly and with honour.
This and dozens of other early Church writings can be found here:
http://www.newadvent.org/fathers/
