Which Roman emperor implemented the worst perscution of Christians?

In terms of death-toll, destruction of Christian property, destruction of Christian sacred texts, the numbers of people imprisoned and tortured to curse Christ and abjure their faith, and how wide-spread it was, the answer is the The Great Persecution of Diocletian, which began in 303, and lasted until 313.

Nero's persecution, by contrast, seems to have been confined to Rome and its surroundings -- and there's precious little evidence that it occurred: Tacitus is the only primary source on the subject, and he only gives it a paragraph. None-the-less, Peter and Paul were both martyred in Rome late in Nero's reign. It was suggested by my old Roman History professor, Alvin Bernstein, that the reason there were no Christian primary sources on the event was because Nero succeeded in killing the all.

But contrary to popular belief, the persecutions weren't conducted on the whim of mad and vicious emperors. Paying divine honors to a crucified miscreant was the height of blasphemy and sacrilege in both Greek and Roman paganism. It violated horrific taboos concerning the dead, not to mention their whole honor-shame/good-luck-vs.-bad-luck theology. (Success was its own vindication; failure and humiliation was proof that it was deserved. Their gods weren't supposed to be more moral than men, but merely immeasurably happier: the fact that Zeus/Jupiter could do all the wretched things he did and still abide in perfect glory and bliss was considered to be proof of, rather than proof against, his deity.)

Also, like it or not, both sides tell us that a major factor was the effectiveness of Christian exorcisms - which included banishing/silencing pagan oracular spirits. The Christians attributed these "victories" to the superior power of Christ over the demons. The pagans said it was that the blessed gods were rightly offended by the mere mention of the name of this crucified Jew, and consequently withdrew in disgust. In their creed, the mere mention of Jesus Christ was "polluting" - it rendered the speaker, the hearers, and the place "unclean."

Unlike today, a crime of this kind was tantamount to high treason because it angered the gods - and when the gods were angry you had floods, earthquakes, famines, military defeats, economic depressions, plagues, and fires that destroyed most of your city.

However, Roman justice being accusatorial, not inquisitorial, and Roman government being notoriously minimalistic, they didn't take pre-emptive steps against people just because their actions might anger the gods someday: they waited for the gods to be angry (as evidenced by some community catastrophe) and then went looking for the albatross-killer in their midst. Thus Justin's jab, "If the Tiber floods, or the Nile doesn't, immediately goes up the cry, 'The Christians to the lion!'"

Persecutions from Nero until Decius (AD 250) were generally local and sporadic, almost all of them Asia Minor, Greece (including the Balkhans), Italy, Alexandria, Marseilles, and the most Hellenized portions of Syria & Palestine. As Berber religion didn't have the same taboos concerning death and the honors paid to dead men, persecutions were virtually unknown in North Africa prior to 250: Christians of that time couldn't think of more than 7 people who had been martyred in all of North Africa before that.

When the Roman dynasties (Julio-Claudians, Flavians, and Antonines) were succeeded by the Syro-Phoenician dynasty of Septimus Severus, and a long string of non-Roman and non-Greek emperors followed, persecutions waned substantially from 193 to 249, as oriental influences forced the old Greco-Roman pantheon and its cult to the periphery and solar religion came to dominate politics and society. There were still minor, local, and often individual-specific persecutions during this time, but nothing like what had been carried out under Domitian and the Antonines. And by all accounts, either by mob-action or the verdict of an old Roman magistrate of the old school.

Things changed with the ascendance of Philip the Arab to the imperial throne. He may have been a Christian himself: though that is disputed by most modern scholars, it is known that he willingly subjected himself to public humiliation and penance before the bishop of Antioch for his part in a massacre.

To the traditionalist Greeks and Romans this was worse than blasphemy: the greatest of all shames in their paradigm was for a man to willingly submit to humiliation, and doing so incurred a curse. For an emperor to do it, brought a curse upon the whole empire. So one of Philip's old-Roman officers, Decius, murdered him, seized the throne, and launched the first systematic, empire-wide attempt to destroy Christianity.

Though a string of short-lived emperors followed for the next 35 years, there was an off-an-on attempt to revive the old Roman state religion, and those attempts match up pretty well with the subsequent persecutions. When Diocletian restored order, he went the furthest - and was the most successful - toward reviving the old Roman state cult, pushing Syro-Phoenician solar religion to the periphery. By that time, however, Christians were ubiquitous, even in the imperial civil service and Diocletian's own household. But when the augurs announced that they couldn't take the auspices because the Christians at court kept crossing themselves (causing the gods and/or their messengers to depart in disgust), Diocletian launched the last and greatest persecution, with characteristic determination and efficiency. But in doing so, he ran into the old problem that martyrdoms not only made more Christians, it made more executed and humiliated miscreants for Christians to defile the empire by honoring them (the cult of the martyrs). The persecution was none-the-less continued by his successors, until Constantine issued his edict of toleration in 311, and persuaded his colleage to do the same 2 years later.

Emperor Nero in 64 CE.

Is it Nero? Or Constantinople ?

That depends on what you mean by worst.

Nero's methods were the cruellest but his persecution was limited largely to Rome. Diocletian's 'Great Persecution' (303-11) was the longest and furthest reaching. Decius was responsible for the first Empire-wide persecution but this lasted for less than two years.