## 1 Introduction
Interestingly, this important indicator has seldom been the subject of quantitative studies based on tick-by-tick data. Only recently has progress in this direction been reported in Ref. [21], where the inter-transaction times, the number of transactions in unit time, and the traded volume have been analysed. Data representing major cryptocurrencies collected from a few trading platforms have shown that the inter-transaction times are long-term autocorrelated with a power-law decay [21] exactly the same as the data from the regular stock and Forex markets [22; 23; 24; 25; 26]. This provides space for future application and testing of relevant stochastic models, such as the Markov switching multifractal duration model [27; 26] and the continuous-time random walk model [28; 29; 30; 31] to the cryptocurrency market data. As the inter-transaction times are directly related to the number of transactions in time unit, the latter quantity was also demonstrated to show long-range power-law autocorrelation [32]. The inter-transaction times and the number of transactions in the stock markets were reported to be multifractal [23; 33], and the same was observed for the cryptocurrency market data [21]. Multiscaling of the related time series has also been reported, with a strong indication that small fluctuations, i.e., the periods of increased trading frequency, show richer multifractality compared to the large fluctuations associated with the periods of less frequent trading [21]. Slower trading thus happens to be more uncorrelated (efficient) than trading associated with a market frenzy. The long-term autocorrelations of inter-transaction times are also responsible for their distribution tail behaviour, which according to the Ref. [21] cannot be approximated by an exponentially decaying function but, contrastingly, in many cases can be approximated by stretched exponential or power-law functions. In the present work, we study data that represent a few quantities that characterise the trading of two major cryptocurrencies: BTC and ETH. These quantities are: the logarithmic price returns, the volume traded in time unit, and the number of transactions in time unit. We investigate the fractal properties of these data both from the univariate perspective, in which the properties of each signal are analysed separately, and from the bivariate perspective, in which we look at the cross-correlations between the respective time series of BTC and ETH. As a particularly novel element, we also study the lagged cross-correlations between these assets and seek a possible asymmetry between BTC\(\rightarrow\)ETH and ETH\(\rightarrow\)BTC directions. Our goal is to answer the following questions:  
1. Do the price returns, the volume, and the number of transactions in the time unit representing the principal cryptocurrencies show any statistical inter-currency cross-correlations that can be detected with the \(q\)-dependent detrended cross-correlation coefficient (defined in Section 2)? 2. Are those cross-correlations, if present, fractal? That is, does the covariance of the fluctuations of these quantities reveal multiscaling/multifractality over a range of scales? 3. Do the cross-correlations, if present, survive if the studied signals have been shifted in time with respect to each other? 4. If so, is it possible to observe any asymmetry between the results with respect to a shift direction (BTC\(\rightarrow\)ETH and ETH\(\rightarrow\)BTC)? 5. What is a proposed explanation for the outcomes? Our paper is organised as follows. In Section 2, we present the data on which our study is based, together with the applied multifractal formalism. In Section 3 we report details of the results, and in Section 4, we sum up the results and discuss their implications.