  The complexity class Quantum Statistical Zero-Knowledge ($\mathsf{QSZK}$)
captures computational difficulties of the time-bounded quantum state testing
problem with respect to the trace distance, known as the Quantum State
Distinguishability Problem (QSDP) introduced by Watrous (FOCS 2002). However,
QSDP is in $\mathsf{QSZK}$ merely within the constant polarizing regime,
similar to its classical counterpart shown by Sahai and Vadhan (JACM 2003) due
to the polarization lemma (error reduction for SDP).
  Recently, Berman, Degwekar, Rothblum, and Vasudevan (TCC 2019) extended the
$\mathsf{SZK}$ containment for SDP beyond the polarizing regime via the
time-bounded distribution testing problems with respect to the triangular
discrimination and the Jensen-Shannon divergence. Our work introduces proper
quantum analogs for these problems by defining quantum counterparts for
triangular discrimination. We investigate whether the quantum analogs behave
similarly to their classical counterparts and examine the limitations of
existing approaches to polarization regarding quantum distances. These new
$\mathsf{QSZK}$-complete problems improve $\mathsf{QSZK}$ containments for QSDP
beyond the polarizing regime and establish a simple $\mathsf{QSZK}$-hardness
for the quantum entropy difference problem (QEDP) defined by Ben-Aroya,
Schwartz, and Ta-Shma (ToC 2010). Furthermore, we prove that QSDP with some
exponentially small errors is in $\mathsf{PP}$, while the same problem without
error is in $\mathsf{NQP}$.
