  Given a classical error-correcting block code, the task of quantum list
decoding is to produce from any quantumly corrupted codeword a short list
containing all messages whose codewords exhibit high "presence" in the
quantumly corrupted codeword. Efficient quantum list decoders have been used to
prove a quantum hardcore property of classical codes. However, the code rates
of all known families of efficiently quantum list-decodable codes are,
unfortunately, too small for other practical applications. To improve those
known code rates, we prove that a specific code family of polynomially small
code rate over a fixed code alphabet, obtained by concatenating generalized
Reed-Solomon codes as outer codes with Hadamard codes as inner codes, has an
efficient quantum list-decoding algorithm if its codewords have relatively high
codeword presence in a given quantumly corrupted codeword. As an immediate
application, we use the quantum list decodability of this code family to solve
a certain form of quantum search problems in polynomial time. When the codeword
presence becomes smaller, in contrast, we show that the quantum list
decodability of generalized Reed-Solomon codes with high confidence is closely
related to the efficient solvability of the following two problems: the noisy
polynomial interpolation problem and the bounded distance vector problem.
Moreover, assuming that NP is not included in BQP, we also prove that no
efficient quantum list decoder exists for the generalized Reed-Solomon codes.
