Uncountable classical and quantum complexity classes

Polynomial–time constant–space quantum Turing machines (QTMs) and logarithmic–space probabilistic Turing machines (PTMs) recognize uncountably many languages with bounded error (Say and Yakaryılmaz 2014, arXiv:1411.7647). In this paper, we investigate more restricted cases for both models to recognize uncountably many languages with bounded error. We show that double logarithmic space is enough for PTMs on unary languages in sweeping reading mode or logarithmic space for one-way head. On unary languages, for quantum models, we obtain middle logarithmic space for counter machines. For binary languages, arbitrary small non-constant space is enough for PTMs even using only counter as memory. For counter machines, when restricted to polynomial time, we can obtain the same result for linear space. For constant–space QTMs, we follow the result for a restricted sweeping head, known as restarting realtime.