This paper has investigated the predictability of the top ten cryptocurrencies’ price dynamics, ranked by their daily market capitalization and trade volume, via the information theory quantifiers. Our analysis considers the Complexity-entropy causality plane to study the temporal evolution of the price of these cryptocurrencies and their respective locations along this 2D map, bearing in mind after and during the Russia-Ukraine war. Moreover, we apply the permutation entropy and the Jensen-Shannon statistical complexity measure to rank these cryptocurrencies similarly to a complexity hierarchy. Our findings reflect that the Russian-Ukraine war affects the informational efficiency of cryptocurrency dynamics. Specifically, the cryptocurrencies notably showed a decrease in informational inefficiency (USD-coin, Binance-USD, BNB, Dogecoin, and XRP). At the same time, the cryptocurrencies with more expressiveness for the financial market, considering the volume traded and the capitalized market, were strongly impacted, presenting an increase in informational inefficiency (Tether, Cardano, Ethereum, and Bitcoin). It clarifies the potential of cryptocurrencies to mitigate exogenous shocks and their capability to use with portfolio selection, risk diversification and herding behaviour.