Mash blackberries in a saucepan with a potato masher. Stir in sugar until juices form; place about 1 tablespoon blackberry juice into a small bowl and stir in cornstarch. Pour cornstarch mixture into saucepan.

Bring berries to a boil, stirring often, until jam is thickened, about 15 minutes. Remove from heat, stir in cinnamon, allspice and lemon juice, allow to cool. Transfer jam to a bowl, cover, and refrigerate until chilled.