The artist utilized the engine's Particle System to create the effects.
According to the artist, the purple spell's straight lines are separate quads with the same material on them. The shape is set to a circle with a small radius. To make them align in a specific shape, he tweaked their pivot and scale. To animate the effect, the artist scaled them on the axis that changes their length.
As for the golden spell, the artist used a similar technique, with the central star being five quads, resized on one axis over time. Stars and rectangles on the outside are also quads created around the circle shape.
You can check out more of Rensei's projects here.