How many Tsar Bomba size nuclear explosions would be required to cause a nuclear winter?

It depends on where they are set off, as nuclear winter is precipitated mostly by the transfer of materials such as smoke and soot from flammable cities (not just nuclear materials) from the blast site into the upper atmosphere.

The Tsar Bomba detonation was scaled back from 100 megatons to 50 megatons to mitigate nuclear fallout, not nuclear winter, and it was not detonated over a city.

It is estimated, from a 2007 study, that one third of the world's arsenal could release 50 Tg (teragrams) of smoke, and reduce global temperatures by -3°C to -4°C for years, with half that shift persisting for more than a decade. There might be more than a -20°C shift in North America, and more than -15°C in Eurasia, covering all agricultural areas.

This would be comparable to the ice age of 18,000 years ago, and would devastate the planet. Even a "small" conflict, involving "50 Hiroshima scale weapons", less than 100 kilotons each, very small in comparison to today's weapons, would be terrible. The study has been criticized, but I believe it to be accurate.