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We study composition series of derived module categories in the sense ofAngeleri H\"ugel, K\"onig & Liu for quasi-hereditary algebras. More precisely,we show that having a composition series with all factors being derivedcategories of vector spaces does not characterise derived categories ofquasi-hereditay algebras. This gives a negative answer to a question of Liu &Yang and the proof also confirms part of a conjecture of Bobi\'nski & Malicki.In another direction, we show that derived categories of quasi-hereditaryalgebras can have composition series with lots of different lengths andcomposition factors. In other words, there is no Jordan-H\"older property forcomposition series of derived categories of quasi-hereditary algebras.

Derived categories of quasi-hereditary algebras and their derived composition series