Diablo – When can I play in Hardcore mode

If you're talking about the Jar of Souls or Matriarch's Bones quests that randomly appear in one of the incorrect defiled crypts, then no. These quests are simply random and can appear on any difficulty or hardcore setting.

It remains to be seen for the rest of the game, I don't know of any specific answers that have been given out by Blizzard staff mentioning hardcore-specific quests.

Since it seems that this question hasn't received a comprehensive answer, here's my best attempt at a full list of all of the differences between Hardcore and Normal (Softcore) modes:

In Hardcore, you:

• Will lose the ability to play your character permanently upon your character's death. You will also have the option to archive a profile of your dead character (playtime, level, appearance, cause of death) in your "Hall of Fallen Heroes" for posterity.

• Will lose all of your character's equipped items, all items in your character's inventory, and all items equipped on your Followers when your character dies. Any items in your Stash, as well as your Gold and Blacksmith/Jeweler status, are retained (and are shared between all of your present and future Hardcore characters).

• Have a separate stash, gold pool, and Blacksmith/Jeweler than your Normal mode characters. There is a dedicated Hardcore-only Auction House, and Blizzard has stated that there will not be a Real-Money Hardcore Auction House. Hardcore mode is an entirely separate ecosystem than Normal mode.

• Cannot trade items back and forth between Hardcore and Normal characters.

• Cannot join a game with non-Hardcore characters (and they cannot join yours).

• Can earn special Achievements for killing bosses and reaching level thresholds on a Hardcore character. Many of these Achievements have special rewards, such as banner designs.

Additionally:

• There are no innate differences in the strength of a Hardcore character versus a Normal character.

• So far, there is no evidence of any difference in the strength of monsters, quality of loot drops, or anything else that affects gameplay aside from the fact that your character cannot be resurrected upon death.