  We introduce several families of quantum fingerprinting protocols to evaluate
the equality function on two $n$-bit strings in the simultaneous message
passing model. The original quantum fingerprinting protocol uses a tensor
product of a small number of $\mathcal{O}(\log n)$-qubit high dimensional
signals [Buhrman et al. 2001], whereas a recently-proposed optical protocol
uses a tensor product of $\mathcal{O}(n)$ single-qubit signals, while
maintaining the $\mathcal{O}(\log n)$ information leakage of the original
protocol [Arrazola and L\"utkenhaus 2014]. We find a family of protocols which
interpolate between the original and optical protocols while maintaining the
$\mathcal{O}(\log n)$ information leakage, thus demonstrating a trade-off
between the number of signals sent and the dimension of each signal.
  There has been interest in experimental realization of the recently-proposed
optical protocol using coherent states [Xu et al. 2015, Guan et al. 2016], but
as the required number of laser pulses grows linearly with the input size $n$,
eventual challenges for the long-time stability of experimental set-ups arise.
We find a coherent state protocol which reduces the number of signals by a
factor $1/2$ while also reducing the information leakage. Our reduction makes
use of a simple modulation scheme in optical phase space, and we find that more
complex modulation schemes are not advantageous. Using a similar technique, we
improve a recently-proposed coherent state protocol for evaluating the
Euclidean distance between two real unit vectors [Kumar et al. 2017] by
reducing the number of signals by a factor $1/2$ and also reducing the
information leakage.
