Hey everyone, these are my own ideas for weekly quests in Destiny 2. Enjoy.
Weekly Crucible Quest "Arena of Light": This quest would be received from Lord Shaxx. You would have to kill 100 Guardians and win 10 matches. The rewards would be Nightfall tier.
Weekly Vanguard Quest "Guardian of Hope": This quest would be received from Commander Zavala. You would have to kill 1000 enemies, defeat 10 Ultra bosses, and complete a Gold tier Public Event. The rewards would also be Nightfall tier.
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https://www.bungie.net/en/Forum/Post/217394818/0/0